Mark Viduka’s 77th-minute strike appeared to have sent Leeds four points clear at the top but second-half substitute Solskjaer headed home a cross from Ryan Giggs a minute from time to save United from a second successive home league defeat.
United almost stole the points at the death but Nigel Martyn pulled off a brilliant save to deny Ruud van Nistelrooy.
The home side create the better chances, especially in the opening 45 minutes but, without injured skipper Roy Keane who is still nursing a knee injury, they lacked the spark to break down a resolute Leeds defence.
United set out with all guns blazing with Martyn forced into smart saves from Beckham and Giggs in the first three minutes.
Van Nistelrooy and Nicky Butt, making a rare start in midfield, also wasted chances in the first half-hour before Leeds threatened for the first time when Harry Kewell played Viduka through on goal.
The Australian striker had plenty of time to pick his spot but fired his shot wide.
United almost broke the deadlock when Beckham struck the post following a swift counter-attack.
Juan Veron, van Nistelrooy and Giggs were all involved before the Welsh winger sent a cross into the middle which the England captain prodded towards goal, only to see his effort strike the right upright.
The match boiled over 10 minutes into the second half when Leeds’ striker Robbie Keane reacted to a late Beckham tackle by pushing him in the face with enough force to knock him over.
Referee Dermot Gallagher showed a yellow card to Beckham and, with United fans calling for Keane’s dismissal, he brandished another yellow much to the Irishman’s relief.
Keane thought he had scored three minutes later when he beat Barthez with a quickly-taken 25 metre free-kick, but Gallagher ruled out the goal having not blown his whistle.
Fabien Barthez then denied Keane following a fine individual run but the visitors went ahead 13 minutes from time.
Kewell and Ian Harte linked well on the left and, from the latter’s cross, Viduka got goalside of Mikael Silvestre to steer a low shot past Barthez from six metres, only his second league goal of the campaign.
United were shocked into action and finally began to pressurise the Leeds defence with Veron pulling the strings in midfield.
But it was Solskjaer, so often United’s saviour from the bench, who got United back on terms, outjumping Harte to head past Martyn from six metres out.
United then went for the jugular with Dominic Matteo lucky not to score an own goal and Martyn keeping out van Nistelrooy in stoppage time.
Arsenal can take top spot from Leeds if they win at Sunderland in a match played later on Saturday.
SATURDAY’S RESULTS:
Premier league
Aston Villa 3 Bolton Wanderers 2
Charlton Athletic 0 Liverpool 2
Everton 1 Newcastle United 3
Fulham 2 Southampton 1
Manchester United 1 Leeds United 1
Sunderland 1 Arsenal 1
Tottenham Hotspur 2 Middlesbrough 1
—Reuters