Defoe fired Sunderland ahead on 34 minutes before Anichebe struck twice in the second half to seal a win.
Jermain Defoe scored his 150th Premier League goal on Saturday as Sunderland climbed off the foot of the table with a 3-0 win over fellow strugglers Hull City, in a match held up by floodlight failure.
The former England striker fired Sunderland ahead on 34 minutes at the inappropriately named Stadium of Light before Victor Anichebe struck twice in the second half to seal a second straight win for David Moyes's side.
The result moved the Black Cats above Swansea City, who drew 1-1 at Everton, but they are still three points from safety.
Hull remain in 18th place and have lost seven of their last eight games under Mike Phelan.
Despite being less than a third of the way through the season, the importance of Saturday's clash was clear to both sides, who will likely face a long battle against relegation.
Anichebe shrugged off a rib injury sustained in the 2-1 win at Bournemouth last time out to retain his place in the Sunderland side, while Robert Snodgrass returned for Hull, who handed a Premier League debut to 17-year-old defender Josh Tymon.
Sunderland goalkeeper Jordan Pickford, who had yet to keep a clean sheet this season, turned behind an early shot from Dieumerci Mbokani, with his opposite number David Marshall fortunate not to concede a penalty after clattering Duncan Watmore.
Sam Clucas forced Pickford into another save low down, but Defoe gave Sunderland the lead with a typically clinical finish.
The 34-year-old latched onto a flick-on from Watmore and ghosted past Curtis Davies and Ahmed Elmohamady before guiding past Marshall for his seventh goal of the campaign.
Anichebe almost added a quick-fire second when his effort flashed just wide, while Watmore drew a smart save from Marshall before the match was delayed for 10 minutes at the start of the second half when there was a blackout in the stadium.
When the lights came back on Ryan Mason spurned a chance to equalise after Snodgrass broke away down the left, and Anichebe made them pay by drilling in Sunderland's second on 62 minutes.
It was the first time the Nigeria international had scored in back-to-back league games since September 2012, but Anichebe wasn't finished.
After Davies was denied brilliantly by Pickford and Patrick van Aanholt twice cleared off the line to add to Hull's frustrations, Anichebe powered in the hosts' third on 84 minutes.
The only blot to Sunderland's copy book was the late dismissal of Senegalese defender Papy Djilobodji for a second yellow card.