Bob Bradley's side won for just the second time under their American manager thanks to Gylfi Sigurdsson's penalty and a brace from Fernando Llorente at the Liberty Stadium.
Swansea's second win in their last 14 league games was well-timed for Bradley as the club's owners Jason Levien and Steve Kaplan had flown in from the United States amid speculation the former Egypt coach was in danger of being sacked after just two months in charge.
While Swansea are now one place above the relegation zone, Sunderland hit rock bottom as they paid the price for a lacklustre display that ruined a recent revival including three wins from their previous four matches.
Bradley conceded this week that Swansea's worst start to a season since 1983 meant Sunderland's visit had taken on "cup final" proportions and he responded by making five changes to his starting line-up.
With torrential rain lashing south Wales, Swansea made a slow start and it was Sunderland who should have struck first midway through the half.
Sunderland striker Jermain Defoe had scored 14 goals in his last 22 away matches in the league, including a hat-trick at Swansea last season, and he came close to extending that fine run when he fired narrowly over.
Swansea threatened for the first time when Sigurdsson's free-kick picked out Alfie Mawson at the far post, only for the defender to head just wide.
Sigurdsson provided another dangerous moment from a free-kick and the Iceland midfielder's teasing delivery deserved better than Jordi Amat's off-target header.
David Moyes' side had managed only one clean sheet this season and they shot themselves in the foot again as Sigurdsson put Swansea ahead in the 51st minute.
Sunderland defender Jason Denayer needlessly handled Wayne Routledge's cross and Sigurdsson drove the resulting penalty down the middle for his third goal in his last four appearances.
Sigurdsson, by far Swansea's most influential player, wasn't finished yet and he played a key role in his side's second goal three minutes later with a clever corner that allowed Spanish forward Llorente to sweep home from six yards.
Amat almost made it three for Swansea with a powerful header from Sigurdsson's free-kick, but Jordan Pickford produced a superb save.
It was Llorente who delivered the knockout blow in the 80th minute as the 31-year-old headed in from Jefferson Montero's cross for his fifth goal of the season.