Antonio Rudiger gave the home side the lead in the first half, but it looked as though United would make it two come-from-behind wins on the bounce when Anthony Martial's brace after the interval put them on the verge of a valuable three points.
The result leaves Chelsea one point clear at the top of the table with closest rivals Manchester City and Liverpool both left to play later today, whereas United's poor record at Stamford Bridge continues to leave them still eighth in the standings.
Rudiger had been a doubt for the match after being forced to pull out of the Germany squad through injury, but he recovered in time to start in a Chelsea team which showed two changes - the same number made by United boss Mourinho from their rousing comeback victory over Newcastle United.
The first sight of goal fell to Chelsea when Eden Hazard - who was the victim of some heavy-handed treatment throughout the match - won a free kick right on the edge of the box, only for Willian to curl his effort high and wide of the target.
United had their first chance five minutes later when Romelu Lukaku could not steer his header on target from Luke Shaw's cross, but that proved to be their best opening before half time as they struggled to break through the Chelsea defence.
It was one defensive lapse which proved to be the difference in the opening 45 minutes when Paul Pogba lost Rudiger from a corner, allowing the Chelsea defender to meet Willian's delivery with a firm header which left David de Gea no chance.
Rudiger turned provider 10 minutes later by floating a pass over the defence for an unmarked Marcos Alonso, whose control let him down inside the area when a better first touch would have left him with a golden opportunity.
The remainder of the first half passed by without much action at either end of the field, though, as Chelsea began to develop a measure of control over the game through their superior possession.
The hosts created the first chance of the second half too, when Alvaro Morata - ineffective in the first half - turned smartly inside the box but then failed to get the power behind his shot to beat De Gea.
United did begin to improve shortly after that scare, though, and their equaliser arrived 10 minutes after the restart when a sustained period of pressure resulted in Martial collecting the ball inside the box and firing a clinical finish past Kepa Arrizabalaga, who had denied Juan Mata seconds earlier.
Chelsea could have regained their lead shortly after the hour mark when again United were undone from a set piece, but this time David Luiz could only glance his header wide of the target from Willian's free kick.
De Gea was able to watch that chance float past the far post, but the Spaniard was called into action five minutes later when he got down to parry N'Golo Kante's low drive away from goal.
It proved to be an important stop with United going down the other end and taking the lead just two minutes later, with Mata and Marcus Rashford combining to find Martial, who cut inside before curling his low finish into the bottom corner.
The goal saw Martial join an exclusive club alongside Paul Scholes and Wayne Rooney as United players to have scored multiple goals away to Chelsea in the Premier League, and the visitors could have put the game to bed 10 minutes from time when Kepa thwarted Ander Herrera after the ball had broken to him in a one-on-one situation.
The fourth official indicated six minutes of stoppage time, and Chelsea needed all of those to get their dramatic late equaliser when Luiz kept a Cesar Azpilicueta cross alive at the back post, nodding it into a dangerous area. Rudiger got on the end of it and saw his header well saved by De Gea, but Barkley was on hand to tuck home the rebound and spark wild scenes of celebrations.
One member of Chelsea's staff made a point of celebrating in front of Mourinho, who reacted angrily to spark a melee on the touchline which spilled out onto the pitch, delaying the restart by a couple of minutes.
However, there was no time for a winner from either side as the spoils were shared, with Chelsea now having gone nine games unbeaten at the start of a Premier League season for only the third time ever.
United, meanwhile, fall just short of recording only their second win at Stamford Bridge in their last 17 Premier League visits and remain seven points off the pace - a gap which could grow to nine this afternoon.
CHELSEA (4-3-3): Kepa; Azpilicueta, Luiz, Rudiger, Alonso; Kante, Jorginho, Kovacic (Barkley 96'); Willian (Pedro 73'), Morata (Giroud 79'), Hazard
MAN UTD (4-2-3-1): De Gea; Young, Lindelof, Smalling, Shaw; Matic, Pogba; Rashford (Sanchez 85'), Mata (Herrera 75'), Martial (Pereira 84'); Lukaku