United was reliant on an early goal from England striker Wayne Rooney, playing his first game in three weeks after a knee injury, to squeeze past West Ham at Old Trafford and earn a match against Fulham. Rooney later missed a penalty.
"In the FA Cup anything can happen so we are pleased to get through," veteran United midfielder Ryan Giggs said.
Arsenal left it late to seal a deserved victory at Emirates Stadium, with Jack Wilshere capping a superb display with an 86th-minute winner to knock out a tired-looking Swansea. Second-tier Brighton, the third-round conqueror of Newcastle, is next for Arsene Wenger's team.
"He is a complete midfielder," Wenger said of Wilshere, who recently returned to action after a 14-month injury absence. "He was outstanding tonight but as a team we played well, and created chance after chance."
Chelsea was booed off by a disgruntled home crowd after another worrying home performance, although the European champions looked to be cruising after goals by Demba Ba and Eden Hazard gave them a 2-0 halftime lead at Stamford Bridge.
Substitute Rickie Lambert scored with one of his first touches and Jason Puncheon grabbed the equalizer with 15 minutes remaining, leaving third-place Chelsea 13 points behind leader United with 16 games left.
Since Benitez's controversial appointment in November, Chelsea has won just two of its seven matches at home in all competitions.
"We have to be more clinical when we have chances," Benitez said. "We have to manage the game, score the third goal and finish the game but we did not do it."
Rooney endured a tough festive period on and off the pitch, first picking up an injury in training on Christmas Day and then receiving the news on Jan. 5 that his wife's 14-year-old sister had died after a battle with a rare genetic disorder.
The striker celebrated his ninth-minute goal against West Ham — when he turned in Javier Hernandez's cross — by kissing his hands, looking to the skies and raising his arms as he walked back to the halfway line.
Rooney skied a penalty over the bar in the 78th minute — his 10th failed spot kick for the club — before United was forced to stave off a late rally from the visitors to seal a place in the last 32.
Wilshere spared Arsenal an energy-sapping extra 30 minutes by running onto a lay-off by Olivier Giroud and smashing a fierce volley low into the corner from the edge of the area, ending Swansea's spirited resistance after a flurry of chances were wasted by Arsenal.
"I don't think I'll ever hit another like that," said Wilshere, who was praised by Wenger and Swansea manager Michael Laudrup after yet another impressive display.
"He can be a very, very good player — but don't put the pressure on him that he can be a superstar," Laudrup said. "It's OK to be fascinated by big talents, when they have the potential to be great players. But to be a world-class player, you need more seasons at the highest level."
Theo Walcott was guilty of a host of missed opportunities, hitting the post with a header, having a shot cleared off the line, and squandering a host of other chances in a one-sided second half.
Swansea, which started with star forward Michu on the bench, leads Chelsea 2-0 after the first leg of their League Cup semifinal, so being eliminated from the FA Cup didn't hurt Laudrup too much — especially given Arsenal's dominant display.
"I'm not too disappointed," Laudrup said. "To be fair, they deserved the win."