The Court of Arbitration for Sport’s decision to lift Manchester City’s two-season European ban meant a fifth-placed Premier League finish would not be enough to seal a return to the continent’s top table.
United looked set to move up to third after Marcus Rashford and Anthony Martial turned things around following Stuart Armstrong’s early opener, only for Obafemi to direct home at the death to seal a 2-2 draw.
Ole Gunnar Solskjaer’s side remain fifth after blowing the chance to move ahead of Chelsea and Leicester on Monday night, when the frustration was clear on the face of the Norwegian and his players.
Ralph Hasenhuttl’s men deserved a share of the spoils, though.
Their relentless pressure forced Paul Pogba into an early error that Armstrong capitalised on.
But United produced a response that the watching Sir Alex Ferguson would have been proud of, with Rashford equalising in the 20th minute and Martial turning things around with a superb strike three minutes later.
Southampton midfielder Oriol Romeu was fortunate to avoid punishment for a foul on Mason Greenwood in a first half that made way for cagey second period, with David De Gea producing a superb stop to deny Nathan Redmond late on.
United looked set to hold on, only for Obafemi to get to a corner before Victor Lindelof to snatch a late point for Saints.
Both managers stuck with unchanged sides having impressed since the restart and Saints started on the front foot in the north west.
But early poise in possession veered into over-confidence when James Ward-Prowse uncharacteristically lost control, with Martial pouncing on the loose ball and forcing Alex McCarthy into a superb one-handed save.
That chance was against the run of play as United were penned back and given little time to breathe, with the visitors’ constant pressing bringing them an opener inside 12 minutes.
Danny Ings caught Pogba in possession just outside his own area and the ball fell to Redmond, who clipped a lovely cross over to Armstrong, who took a touch and slammed home at the far post.
It was a nightmare start for United but, just like their last home match against Bournemouth, they quickly turned things around.
While Rashford was rightly adjudged offside before turning home a Luke Shaw cross, he would find a 20th-minute leveller. Pogba’s cross into the box was held up under pressure by Martial, who managed to prod through for the England forward to strike low past McCarthy.
And, by the 23rd minute, United had turned things around.
Martial took possession from Bruno Fernandes on the left and cut inside past Kyle Walker-Peters before unleashing a thumping strike into the top left-hand corner.
McCarthy prevented Pogba adding a third from distance, but the drinks break had taken the sting out of a helter-skelter opening.
The main talking point after that was Romeu’s challenge on Greenwood, which somehow avoided punishment by VAR.
Saints returned from the break with renewed hunger as Redmond and Ings looked to draw them level, with Armstrong seeing penalty appeals ignored.
Rashford went close with a dipping shot at the other end, but Solskjaer’s frustration at their inability to finish the game off was underlined by a furious bellow at Fernandes.
Rashford saw an effort somehow blocked from point-blank range, with Shaw limping off before Martial blazed into the Stretford End after a slaloming run.
But Saints were continuing to ask questions, posing problems down the flanks and testing United’s resolve.
De Gea forced a wonderful fingertip stop to thwart Redmond before a sickening collision between Walker-Peters and Brandon Williams led to a lengthy break in play.
But instead of ending in a whimper, the game provided late drama as Obafemi turned home a stoppage-time corner to the disbelief of the United contingent.