Runaway leaders Manchester City, who're in action in the EFL Cup tonight, remain 10 points clear at the top, needing just six more to confirm their 2020/21 title.
Neither side tonight created many clear opportunities, with United's best chance only coming from a free-kick by Marcus Rashford which was deflected over the bar by goalkeeper Illan Meslier.
The introduction of Edinson Cavani couldn't help inspire a late win either, while Stuart Dallas and Mateusz Klich were denied at the other end by Dean Henderson.
Yet the Peacocks climbed above Arsenal in ninth place, boosting their ambitions of finishing in the top-half, while Manchester United remained rooted in second.
Player ratings for Manchester United
Dean Henderson - 7/10
Much to his respite, it was a calm afternoon for him, and he just had to make a few straight-forward saves. Another clean sheet in the bag for Manchester United's custodian.
Aaron Wan-Bissaka - 7/10
He made his presence felt at both ends for Manchester United, coping well with Jack Harrison's threat, whilst also going forward to create one good chance for Bruno Fernandes. Wan-Bissaka completed five of his six dribbles and made five interceptions in the match, too.
Victor Lindelof - 7/10
Lindelof remained calm and composed when going against Patrick Bamford, who couldn't find a way past him. The Swede made one bad pass in the first-half which caused a nervy moment. But otherwise, it was a good performance from him.
Harry Maguire - 7/10
The England international held his ground well and kept everything sorted at the back. He even had time to get the ball forward a few times, which he did to some effect.
Luke Shaw - 6.5/10
Shaw was let off the hook for a handball inside the area, so he got really lucky there. But performance-wise, the Manchester United left-back was impressive. He linked-up with Rashford well.
Scott McTominay - 6/10
McTominay was a pale shadow of his ferocious self that ripped Leeds to shreds in the first leg. The spark with which he scored two goals was sorely lacking here and offered nothing in the attack.
Fred - 7/10
He was always eager to get involved in the attack and laid some good throughballs too, including one for Mason Greenwood which created a nice chance.
Daniel James - 7/10
He ran at Leeds with aggression and caused problems with his movements, winning a free-kick which Rashford nearly scored from. James had one good chance to score just moments before he was taken off, but the ball was snuffed out at the last second.
Bruno Fernandes - 6/10
Another insipid performance from the Portuguese talisman, who hasn't looked like his former self for a while now. A cause of concern for Solskjaer as Manchester United seek the Europa League title.
Marcus Rashford - 7/10
A lively performance from Rashford, who constantly took the game to Leeds with a high press, and forced goalkeeper Illan Meslier into some good saves, including from a well-taken free-kick in the first-half.
Mason Greenwood - 6/10
Ole Gunnar Solksjaer said last week that Greenwood is not ready for the centre-forward role yet. He proved to be correct. Manchester United should just stick with Cavani for now.
Paul Pogba - 6/10
He was at the receiving end of a half-chance but squandered it after failing to control the ball.
Edinson Cavani - N/A
Solksjaer subbed him on to try and find a late winner, but it was too little, too late.
Donny van de Beek - N/A
He's almost a forgotten figure in the Manchester United squad and made an anonymous cameo appearance.