Cardiff are now 11/4 to stay up after they dragged struggling Brighton into a real relegation dogfight with just a handful of Premier League games left.
Neil Warnock declared Cardiff's Premier League survival bid "alive and kicking" after securing a vital 2-0 win at fellow strugglers Brighton.
The 18th-placed Bluebirds arrived at the Amex Stadium five points adrift of their opponents, but their 2-0 brought them within two points with both sides facing a tough run-in.
Brighton have a game in hand, but with three top six sides to face they'll have their work cut out, especially since they've lost six on the bounce. Cardiff also have a tough run starting with Liverpool on Sunday.
Premier League relegation run-in
Brighton: Wolves (A), Tottenham (A), Newcastle (H), Arsenal (A), Man City (H)
Cardiff: Liverpool (H), Fulham (A), Crystal Palace (H), Man Utd (A)
Despite facing title-chasing Liverpool on Sunday and finishing the season at Manchester United, City manager Warnock retains hope of pulling off an unlikely great escape.
"There were so many pundits this morning said it would be our last game in the Premier League. We're alive and kicking," said Warnock.
"I was proud of the performance: it was disciplined, I think we could have created a few more (chances) but overall I was delighted with all the departments.
"I've never lost belief that we can win games. But whether we can get enough points is a different ball game. I think we were 100-1 in something today (to be relegated) - 100-1 on that was. There's a lot of water to go.
"At least now it makes Sunday's game more important. It's a great occasion Sunday, full house against one of the best teams in the world, and we aren't relegated yet."
No points target in mind
Warnock admits he does not have a points target in mind to beat the drop.
"No, we don't. We might not get another point. I honestly don't know," he added. "But we're quite capable of picking points up off any of the teams we've got to play.
"You might think I'm in cuckoo land, but I think that we've got that ability on our day."
Brighton's fourth successive league defeat without scoring leaves them anxiously looking over their shoulders going into a tough run-in which includes games against Tottenham, Arsenal and Manchester City.
After making a bright start to the match, the Seagulls failed to bounce back from being thrashed 5-0 by Bournemouth on Saturday and were jeered off by the home crowd.
Albion manager Chris Hughton admits his under-performing players are suffering a crisis of confidence but feels they can recover to retain their top-flight status.
"At the minute, it's a big blow and a bad week," said Hughton.
"I do believe we will stay up because I have to. It's very much in our own hands.
"It's been a really bad week, and it's not a nice feeling at the moment, but we have to use that in whichever way that we can because we need to make sure we get enough points to stay in this division.
"Confidence is an issue, yes. That's normal. When you're not winning games, it knocks your confidence."
Seagulls striker Glenn Murray headed against a post in stoppage time, while Lewis Dunk was denied a first-half penalty after appearing to be hauled down by Bruno Ecuele Manga.
Hughton added: "At the time I wasn't sure, but I've seen it since. It's a definite penalty."