Lukaku's first goal was cancelled out by a magnificent set piece from James Ward-Prowse, but Lukaku struck in the closing minutes to secure a hard-fought three points for his side.
United started brightly in the North-West and Lukaku came close to opening the scoring as early as the third minute, with Angus Gunn having to get down well to his left to parry the Belgian's 10-yard effort away from goal.
The home side continued to pepper the Saints' goal with Gunn having to make a timely block from Lukaku, but the visitors managed to create an opportunity of their own with Jannik Vestergaard missing the target with a header.
However, the chance helped settle Southampton down, and after easing their way into the contest they took the lead in stunning style as Valery found the top corner of David de Gea's net from 25 yards to register his first goal for the South-coast outfit.
Going behind sparked United back into life, but chances continued to be squandered by Lukaku, who miscued an effort from 10 yards with his weaker right foot and then glanced a header wide of the target from close range.
Chris Smalling went even closer to a leveller, with the defender hitting the far post with a 12-yard header after meeting an outswinging corner.
United were dealt a blow soon after the restart when Alexis Sanchez sustained what could prove to be a significant knee injury, but the home side levelled the game moments afterwards when Pereira curled a pinpoint strike into the top corner from 25 yards out.
Southampton continued to look for goals at the other end, but United completed the turnaround just six minutes later when Lukaku switched the ball onto his right foot inside the area before firing a low effort into the bottom corner.
The goals were harsh on the visitors, but they did not prevent the Saints going in search of an equaliser, although Pierre-Emile Hojbjerg could only send a shot from the edge of the area wildly over the bar.
However, Southampton eventually did get back on level terms with their second tremendous goal of the game as Ward-Prowse curled a free kick into the top corner of De Gea's net.
Southampton showed the greater ambition during the closing stages and forced a couple of corners, but United always posed a threat when they went forward and it came as little surprise when a fifth and decisive goal was scored.
The ball broke kindly for Lukaku on the edge of the area, and the forward proceeded to send a low first-time effort into the bottom corner of Gunn's net from 20 yards out.
MANCHESTER UNITED (4-2-3-1): De Gea; Young, Smalling, Lindelof, Shaw; McTominay, Pereira (Fred 82'); Sanchez (Dalot 52'), Pogba, Rashford (Chong 95'); Lukaku
SOUTHAMPTON (3-5-1-1): Gunn; Bednarek, Yoshida (Gallagher 90'), Vestergaard; Valery, Ward-Prowse, Romeu, Hojbjerg, Bertrand; Redmond; Austin (Armstrong 63')