Matty Fryatt's hat-trick earned Hull a 3-1 win and kept them in with a fighting chance of reaching the play-offs while Barnsley can finally put their relegation fears to bed despite an eighth defeat in 10 npower Championship matches.
Nick Barmby's side could still catch fifth-placed Blackpool or Cardiff who are just below them if they win their last two matches and either one of those two teams fails to take another point. However, both have a superior goal difference over the Tigers, who also trail sixth-placed Middlesbrough by two points.
The breakthrough came eight minutes before the break as Cameron Stewart's low cross found Fryatt and he stabbed home from close range as the visiting defence appealed for offside.
Fryatt netted his second goal on the stroke of half-time, Richard Garcia providing a delivery which needed only a touch to turn in.
The visitors scored just their third goal in 10 matches, Jim McNulty ghosting across to the near post to nod in Andy Gray's corner with 16 minutes to play.
A flurry of corners offered Barnsley some hope of a late leveller but City put the game to bed when Fryatt jinked past two players and coolly found the bottom corner to cap a superb display in the 83rd minute.