Speaking back in early December, a fortnight before he was sacked by executive vice-chairman Ed Woodward, Mourinho claimed that United required 'a minor miracle' if they were to finish in the top four.
It seems that all was actually needed for United to be considered among the division's elite once more was a new man to guide them forward, as Solskjaer has collected 29 points from a possible 33 to lift his side into second place.
A home draw with bitter rivals Liverpool last weekend allowed Arsenal to pounce, with the Gunners overtaking United in the fourth and final Champions League qualification spot, but the Red Devils are just a point worse off and travel to the Emirates Stadium in a little over a week's time - a ground at which they have already won during Solskjaer's tenure.
Even a point at home to league leaders Liverpool has to be considered a positive outcome for the Red Devils, having lost three players to injury in the first half - four if you include Marcus Rashford, who could barely run after being clattered by Jordan Henderson - keeping them ticking along in their pursuit of a top-four finish.
The pressure was back on in midweek for the trip to Crystal Palace, a test they passed with flying colours as Romelu Lukaku struck twice to end his barren run in front of goal and Ashley Young added a third as United prevailed 3-1 in the capital.
Even without eight first-team stars, Solskjaer still managed to get the best out of his squad and the interim manager is now the overwhelming favourite to land the job on a permanent basis. United are still in the Champions League and FA Cup, but there is a sense that finishing above Arsenal and Chelsea in fourth will be enough to make the decision a formality.
No matter what way you look at it, Solskjaer has worked - to use that phrase from Mourinho - a minor miracle at Old Trafford. United have won at least five more Premier League points than any other side (29) since the iconic striker was appointed a week before Christmas, enough to suggest that they could even be ready to challenge for the title again next term with one or two additions.
Challenging Liverpool and Manchester City at the summit is out of the question over the remaining two-and-a-half months of the campaign, but Solskjaer and his players still have plenty to aim for. There is the small matter of the Champions League last-16 second leg against Paris Saint-Germain next week, before trips to Arsenal in the Premier League and Wolverhampton Wanderers for a place in the semi-finals of the FA Cup.
United are certainly not lacking momentum ahead of this defining period in their campaign, and three more points on Saturday will set them up nicely for what could be a memorable conclusion to the season.
Recent form in Premier League: WDWWDW
Recent form (all competitions): WWLWDW
Ralph Hasenhuttl fell short of describing Wednesday's relegation six-pointer with Fulham as a must-win game, instead labelling it a "should-win game". In the eyes of supporters, though, only victory would do.
Southampton were not exactly a their best, but first-half goals from Oriol Romeu and James Ward-Prowse - both coming after some questionable goalkeeping from Sergio Rico - ensured that the three points went the Saints' way in a 2-0 win.
The victory, a fifth in 15 matches for Hasenhuttl in all competitions, was enough to lift the south coast side above Cardiff City into 17th place. Southampton are far from safe with 10 matches to go, but they are above the dotted line for now.
A tough run awaits Southampton, though, as they follow up this trip to Old Trafford - a ground they have actually tended to perform well at in recent years - with a home match against Tottenham Hotspur on March 9.
Southampton have won just one of their last 33 Premier League games against big-six opponents, losing 23 of those, though that solitary victory did come just a couple of months ago against Arsenal.
The 3-2 win over Arsenal in mid-December has helped to kickstart the Saints' campaign, with wins over Huddersfield Town, Leicester City, Everton and Fulham following, as well as a well-earned goalless draw at Chelsea.
Another major scalp this weekend would potentially be enough to lift Southampton up to 15th place, though defeat, coupled with victory for Cardiff in their meeting with Wolverhampton Wanderers, would see them drop back into the relegation zone.
Hasenhuttl admitted after the win over Fulham that there is 'no margin for error' between now and the end of the season. Come through this difficult run, which also includes a home meeting with Liverpool in a month's time, out of the bottom three, and the Austrian will be on the verge of pulling off the one job he was brought in to do - keeping Southampton in the Premier League.
Recent form in Premier League: WDDLLW
Rashford and Marcos Rojo recovered from injury to make the bench for United's midweek trip to Selhurst Park and both are in contention to play a part on Saturday.
Fringe players Alexis Sanchez and Fred did not offer a lot on their rare starts last time out to force their way into Solskjaer's plans, and it is expected that one or both men will make way here, depending on who else is available.
Question marks remain over the fitness of a number of players, but either way Solskjaer is unlikely to be able to rotate his side as much as he would have liked ahead of the trips to PSG and Arsenal.
One man who appears certain to start is striker Lukaku, who has been involved in nine goals in his last 10 Premier League appearances against Southampton (seven goals, two assists), though both his goals against the Saints for United have been scored away from home.
Danny Ings has an outside chance of returning to fitness in time for Saturday's match, but Hasenhuttl will surely be tempted to stick with Nathan Redmond - his side's liveliest player against Fulham - in the forward line, possibly joined by Stuart Armstrong after he was left out in midweek.
Following his close-range finish against Fulham, Ward-Prowse has now scored four Premier League goals this season - he has never scored more in a single campaign in the competition.
Manchester United possible starting lineup:
De Gea; Young, Smalling, Lindelof, Shaw; Pereira, McTominay, Pogba; Alexis, Lukaku, Rashford
Southampton possible starting lineup:
Gunn; Bednarek, Yoshida, Vestergaard; Valery, Ward-Prowse, Romeu, Hojbjerg, Bertrand; Armstrong, Redmond
Head To Head
Man United have won only one of their five home Premier League games against Southampton since Sir Alex Ferguson retired, having previously won 13 out of 14.
The Saints have drawn their last two matches against the Red Devils, including a 2-2 draw in the reverse fixture after leading 2-0 - they have not avoided defeat in three consecutively against them since May 1989.
In the history of the Premier League, United have come from behind to win eight matches against Southampton - a joint-record in the competition.
We say: Manchester United 2-0 Southampton
Even accounting for their mounting injury concerns, plus their underwhelming recent return at home to Southampton, United are on a roll and should have enough to add three more points to their tally this weekend.