The Cottagers took a shock lead through Steve Sidwell but were unable to keep United out as they fell under ever-increasing pressure and were beaten twice in the space of as many minutes by Robin van Persie and Michael Carrick.
Rene Meulensteen's men refused to throw in the towel, though, and grabbed a stoppage-time leveller through Darren Bent to pile more pressure on United boss David Moyes.
Emmanuel Adebayor's second-half strike gave Tottenham a 1-0 Premier League home win over Everton on Sunday.
The two top-four chasers played out a goalless draw at Goodison Park in early November but on this occasion, Spurs' Togo international frontman was able to make the breakthrough when he brought down Kyle Walker's chipped free-kick in the 65th minute and fired past Tim Howard.
The victory at White Hart Lane moved Tim Sherwood's side above the Toffees into fifth spot, three points behind fourth-placed Liverpool.
Everton were the better side in the first half and should have taken the lead through Leon Osman. The midfielder was presented with several chances but couldn't find a way past Hugo Lloris.
After the break, Spurs came into the contest and took the lead with their first shot on target. Walker lofted a quickly-taken free-kick up to the edge of the box where Adebayor held off Seamus Coleman before drilling the ball low past Howard.
Everton pressed for an equaliser and had a decent shout for a late penalty turned down when Etienne Capoue appeared to trip Coleman inside the box.
Both clubs are back in league action on Wednesday night. Everton host Crystal Palace, while Spurs travel north to face Newcastle.