Hull boss Steve Bruce was not ready to declare his side safe from relegation despite admitting it would take a "huge turnaround" for the Tigers to go down.
City ensured they would head to London for their FA Cup semi-final next weekend with the comfort of a nine-point gap over the bottom three following a 1-0 home win over Swansea.
George Boyd's first-half header was enough to settle a bitty contest and, more significantly, realise Bruce's target of 10 wins for the season.
The Tigers have now accumulated 36 points - more than they managed when staying up in 2008/09 - yet despite the feel-good factor around the KC Stadium, Bruce remains cautious of forecasting another year in the top-flight.
The 53-year-old pointed at Fulham's surprise 2-1 win at Aston Villa on Saturday as proof that nothing is yet decided, while a win for West Brom at Norwich could also prove vital in the survival scrap.
"I've said all along that we need 10 wins," he said.
"We've gone a long way towards being safe, but it is ridiculous - all of a sudden that result for Fulham and West Brom winning at Norwich has concertinaed the whole thing again.
"When you look at Fulham's three home games they'll be thinking they've still got a squeak. It's still all to play for."
Hull can also bank on the advantage of a comparatively healthy goal difference to those below them, although they do face a tough run-in with Manchester United, Arsenal and Everton still lying in wait.
"You could say we are 10 (points clear) with goal difference, but it's not over with five to go," Bruce added.
"It will take a huge turnaround (for Hull to go down), but we can't take anything for granted."
Bruce was particularly pleased for match-winner Boyd, who played his first game since his ban for spitting at Manchester City goalkeeper Joe Hart.
It is a charge Bruce labelled "ridiculous", although he joked Boyd had earned some luck back after a rare header for the winner.
"For Boydy in particular I'm delighted," he said.
"It was a ridiculous decision by the FA to ban him. But, no disrespect to Boydy, you know it's your day if he scores with a header."
Hull's win owed more to simple fortune, though, as they had to defend grimly after losing their grip on midfield in the second half.
"I've said for the last two or three weeks it's not about performances," Bruce said.
"Their resilience is there for everyone to see again, and the reason we are where we are is they have a great attitude and are determined to see it through. Thankfully we've done it."
Swansea boss Garry Monk was at a loss to explain his side's poor performance after he had stuck with the same starting XI that thumped Norwich 3-0 last weekend.
The rookie manager singled out defender Angel Rangel for his part in the goal after the Spaniard was easily out-jumped by Boyd.
"I thought it was poor defending in the build-up to the goal and then Angel has to do much better at the back post," he said.
"You have to be stronger than that. You have to be willing to get hurt.
"That's what you have to do at the back post - you know it's coming. He knew that cross was coming, he had time to think. It's not like it's a split-second.
"We have been working all week on those sorts of situations, so that's the disappointing part for myself."
Swansea made a vast improvement after the break, but a lack of cutting edge meant they failed to take anything as Hull appeared guilty of letting their minds wander to next weekend's trip to Wembley against Sheffield United.
"It's not what I expected," Monk said.
"In the second half we were marginally the better team. We pushed a little bit, we were better than what we were in the first half, but we didn't really create enough. It was all half-chances for both teams.
"Did we do enough? Maybe to get a draw but I can't grumble."