Cardiff kept in touch with the Premier League strugglers above them by beating West Ham 2-0.
Junior Hoilett and Victor Camarasa scored as Cardiff ended a run of three successive defeats with a victory that was rarely in doubt from the fourth minute.
But Southampton's comeback win against Tottenham leaves Cardiff in the bottom three and two points from safety.
Although Cardiff remain in the relegation zone with eight games left to play, this victory will offer hope to Neil Warnock's side that they can survive.
Half of those games might be against top-six sides – Chelsea and Liverpool at home and the two Manchester clubs away – but there are enough opportunities to pick up points from sides around them.
What they are unlikely to face, however, are opponents as insipid as West Ham.
The Hammers might have won their last two games, but travel sickness has set in and Manuel Pellegrini's side have now lost five of their last six on the road without scoring.
West Ham's fate was sealed from the fourth minute when Hoilett finished smartly at the near post.
After holding a half-time lead for only the second time in 30 their Premier League games, Cardiff's win was secured when Camarasa squeezed home a second seven minutes after the interval.
Talk of a rift in the Cardiff dressing room was put to bed immediately as the Bluebirds' fast start was rewarded with a rare first-half goal.
Lukasz Fabianski had already held Aron Gunnarsson's low shot before Camarasa surged through the Hammers midfield.
Camarasa freed Josh Murphy and his cross was firmly finished at the near post as Hoilett claimed his third goal of the season.
Oumar Niasse was stopped by a timely Issa Diop tackle and chipped over a gaping goal from an acute angle as Cardiff continued to press.
West Ham's play lacked any intensity with their two playmakers, Felipe Anderson and Manuel Lainzini, too deep to make any impression on the contest.
Niasse squeezed another half-chance just wide after Sean Morrison had flicked on Gunnarsson's long throw before the Hammers belatedly stirred just before the break.
Ryan Fredericks' pace stretched Cardiff and Morrison diverted the ball into the path of Robert Snodgrass, who was unable to keep his effort down from 14 yards.
Pellegrini swapped Anderson for Marko Arnautovic at half-time, but the visitors were as lethargic at the start of the second period as they had been in the first.
Cardiff had already worked a couple of promising positions before Murphy nodded Hoilett's cross goalwards.
Camarasa was played onside by Fredericks and the Spaniard bundled the ball between the post and Fabianski from two yards out.
Niasse passed up two great chances to extend Cardiff's lead, Fabianski excelling himself from the second with a superb double save to also deny Camarasa.
The Hammers had barely mustered a shot in anger, but Declan Rice strode forward from his midfield station 20 minutes from time to strike a post from 25 yards.
Arnautovic then forced Neil Etheridge into a full-length save with a powerful attempt, but Cardiff eased to a win that gives them renewed hope.
