Meanwhile, there were huge victories for Sunderland and West Ham over fellow relegation rivals, leaving the battle to avoid the drop up for grabs with the bottom 11 teams separated by just six points.
Eden Hazard and Fernando Torres scored second-half goals to seal a fourth straight victory for Chelsea, which climbed above Arsenal and Manchester City in the standings ahead of their games on Monday and Sunday respectively.
"Now we're top of the league for the first time in a long time and hopefully we can stay there," Chelsea goalkeeper Petr Cech said.
United was on the brink of a crisis, by its lofty standards, after losing three games in six days to kick off 2014, but a 2-0 win over Swansea will boost the confidence among David Moyes' squad and keeps his seventh-place team in touch with the top four.
Antonio Valencia and Danny Welbeck scored in a 13-minute span at the start of the second half for United, which was inspired by another brilliant display by 18-year-old winger Adnan Januazaj.
"His natural talent and ability is up there with the best," Moyes said.
Everton will spend the night in fourth thanks to a 2-0 win over Norwich, and Tottenham also boosted its chances of finishing in the Champions League spots by earning a 2-0 victory over Crystal Palace, which slipped to last place.
That's because Sunderland, which started the day at the bottom, beat Fulham 4-1 mainly thanks to a hat trick by winger Adam Johnson. West Ham was next to last at kickoff but climbed out of the relegation zone with a 2-0 win at Cardiff, despite playing the final stages with 10 men following James Tomkins' sending-off.
Carlton Cole and Mark Noble were the scorers for West Ham, earning their under-fire manager Sam Allardyce some breathing space after a dreadful run of form and condemning Cardiff coach Ole Gunnar Solskjaer to a defeat in his first league match in charge.
England midfielder Adam Lallana scored Southampton's winner in a 1-0 victory over West Bromwich Albion in the day's other game.
Chelsea manager Jose Mourinho said this week it wouldn't even entertain bids of 100 million pounds ($165 million) for Hazard, amid reported interest from Paris Saint-Germain, and it is easy to see why.
The Belgium winger is Chelsea's top scorer this season with nine goals and he underlined his increasing value to the Blues by ending Hull's stubborn resistance with a well-taken individual goal, twice cutting inside before delivering an emphatic finish.
"He is not a talented kid anymore," Mourinho said. "He is now more than that. He is a player who feels and accepts the responsibility, feels and accepts that a team player is more than a talented player and he has done it without losing his fantastic ability and fantastic talent."
Torres, whose best form appears to be returning, marked his 200th Premier League appearance with the second goal three minutes from time and it was also a good day for Cech, who earned a club-record 209th clean sheet.
Behind the top three, who are favoured to battle it out for the title, the race for Champions League spots is heating up.
By beating Swansea, United did more than just avoid a fourth straight defeat for the first time since 1961. It breathed new life into the team's faltering league campaign and provided some welcome respite for Moyes, whose position has been under increasing scrutiny.
"It's the same size as any other. You only get three points," Moyes said of the win, but the reaction inside Old Trafford to Valencia's 47th-minute opener from close range was one of relief more than joy.
Welbeck then glanced home Patrice Evra's scuffed shot in the 59th for his sixth goal in as many league games, ensuring United remained nine points behind Chelsea.
Gareth Barry and Kevin Mirallas scored Everton's goals at Goodison Park to leave Roberto Martinez's in-form team five points behind Chelsea and Norwich without a win in six league matches.
Spurs rode their luck against Palace early on, with Jason Puncheon blazing a first-half penalty way over the bar, before Denmark playmaker Christian Eriksen — enjoying a new lease of life under recently hired manager Tim Sherwood — and Jermain Defoe scored in the second half for the hosts.
It is the start of a long goodbye for Defoe after his move to MLS side Toronto at the end of February was confirmed on Friday.