The champions looked dead and buried but two penalties from Wayne Rooney and a late equaliser from Javier Hernandez silenced the home fans.
United, who missed the chance to draw level with neighbours Manchester City at the top of the Premier League, bossed the early exchanges and twice had penalty shouts denied by referee Howard Webb.
But Chelsea opened the scoring nine minutes before the break. Daniel Sturridge turned Patrice Evra before wriggling along the byline and sending in a cross that bounced in off Jonny Evans' chest.
Sir Alex Ferguson had mentioned United's freezing dressing room before the game and his side were caught cold at the start of the second half.
Fernando Torres' pinpoint cross found fellow Spaniard Juan Mata just outside the six-yard box and he smashed a brilliant first-time volley into the net.
It went from bad to worse for United soon after when David Luiz rose unmarked and his header bounced in off Rio Ferdinand's shoulder.
The Red Devils pulled one back when Sturridge barged into Evra in the Chelsea box and Rooney blasted in the spot-kick.
And it was game on midway through the second half when the striker doubled his tally after Branislav Ivanovic was adjudged to have tripped Danny Welbeck.
Chelsea looked as if they were going to hold on to put some daylight between themselves and their rivals for fourth place, but with six minutes left substitute Hernandez rose unmarked to head home Ryan Giggs' cross.
Mata nearly won it for the Blues at the death with a fine free-kick that was brilliantly turned away by the much-maligned David De Gea.
In Sunday's other game, Papiss Demba Cisse announced his arrival at Newcastle with a stunning strike to earn his new side a 2-1 win over Aston Villa at the Sports Direct Arena.
Robbie Keane cancelled out Demba Ba's 30th-minute opener in first-half stoppage time but debutant Cisse secure three points for the Magpies with a thunderous half-volley.
The Senegal striker chested down Jonas Gutierrez's cross and found the top left corner with a sweetly-struck shot 19 minutes from time.
Cisse could have had a second in the dying stages had Carlos Cuellar not got his studs to another well-taken volley from the Toon new-boy.
Magpies goalkeeper Tim Krul had kept his side in the contest early on, twice denying Darren Bent and, on the first occasion, recovering quickly to save from Stilyan Petrov on the follow-up.
That set the stage for Ba to sweep in Ryan Taylor's cross on the half-hour mark, although the provider was injured in the process and was stretchered off after a lengthy delay.
Villa went close to an equaliser when Stephen Ireland curled a shot narrowly wide while Cisse, Davide Santon and Ba all had chances to stretch the hosts' lead.
But Keane sent the teams into the break on level terms when he volleyed in Charles N'Zogbia's cross with the last kick of the half.
Cisse wasted a gilt-edged headed chance before his strike while Krul pulled off a point-blank save to keep out Gary Gardner's header as Villa pushed for another equaliser at the death.
Manchester City opened up a three-point gap at the top of the table after beating Fulham 3-0 while Arsenal recorded their first win of the year with a 7-1 win over Blackburn.
Sergio Aguero opened the scoring for Manchester City from the penalty spot after Adam Johnson had been fouled, then Fulham defender Chris Baird doubled the lead when he turned Johnson's low cross into his own goal, before Edin Dzeko wrapped up the win 18 minutes from time.
Elsewhere, Thierry Henry scored his first Premier League goal since his return to Arsenal in their 7-1 drubbing of Blackburn.
It was a miserable afternoon for Rovers, who also saw defender Gael Givet shown a first-half red card for a two-footed challenge on Robin van Persie.
The Holland striker began Arsenal's goal frenzy in the lunchtime kick-off after just two minutes, hitting his second seven minutes before half-time and completing the hat-trick in the 62nd minute.
Morten Gamst Pedersen had equalised for the visitors but Alex Oxlade-Chamberlain (two) and Mikel Arteta redressed the balance before on-loan Henry, on as a substitute, scored in injury time.
Sunderland's impressive form under Martin O'Neill continued as a strike from James McClean secured them a 1-0 win at 10-man Stoke.
In another contest where snow fell throughout, a first half of few chances finished with Potters defender Robert Huth sent off for a tackle on David Meyler.
McClean then netted on the hour, bursting through the hosts' defence and tucking the ball away to seal a victory which means the Black Cats have now taken 22 points from the 30 on offer since O'Neill took over.
Swansea's on-loan midfielder Gylfi Sigurdsson played a major role in Brendan Rodgers' side coming from behind to register only their second away win of the season at West Brom.
The Iceland international, who joined from Hoffenheim last month until the end of the season, cancelled out Marc-Antoine Fortune's opener and laid on the winner in a 2-1 win for Danny Graham.
Second-half goals from Andrew Surman and Anthony Pilkington ensured Norwich ended a nervy afternoon with a 2-0 win over strugglers Bolton at Carrow Road.
The game looked set to peter out into a dull goalless draw until Surman volleyed home in the 70th minute and Pilkington wrapped up the win just before the end with an instinctive finish after Bolton goalkeeper Adam Bogdan had parried Russell Martin's drive.
Substitute Kevin Doyle inspired a second-half Wolves comeback at QPR to secure a 2-1 victory.
The Republic of Ireland international came off the bench at half-time and helped set up Matt Jarvis' equaliser 51 seconds later, before going on to net the winner.
It had looked like QPR boss Mark Hughes would be going home with the three points after debutant Bobby Zamora fired Rangers into a deserved lead just days after his move from west London rivals Fulham.
Wolves were given a way back into the game, though, when fellow striker Djibril Cisse was shown a straight red card for an altercation with Roger Johnson in the 34th minute.
Wigan benefited from one of the more bizarre goals the league will see this season but had to settle for a 1-1 draw against Everton at the DW Stadium.
It had been a thoroughly forgettable affair until the 76th minute when Phil Neville blocked a Jean Beausejour cross and the ball somehow found its way past Tim Howard for a hugely embarrassing own goal.
But the hosts' lead lasted only seven minutes, substitute Victor Anichebe heading in the equaliser.
Sunday's Premier League Results
Saturday's Premier League Results