A first-half header from George Boyd saw the Tigers past the visiting Swans on Saturday - with three points lifting them away from their weekend rivals and nine clear of the relegation zone.
With many sides having only five more games to come, that gap could be enough to see Hull safe.
Bruce will not allow his side to take their foot off the gas, though, with there a FA Cup semi-final against Sheffield United to come next weekend and important league games after that as he endeavours to make the 2013/14 campaign as memorable as possible.
He said: "I've said all along that we need 10 wins. Looking at the league as we sit tonight, it's quite a remarkable season, but it's given everybody a lift.
"We've said for the last two or three weeks it's not about performances, but the resilience is there for everybody to see. We are where we are and have a great attitude towards it. We're determined to see it through.
"We've gone a long way to being safe but it's ridiculous - that one result for Fulham, and West Brom winning at Norwich, has just concertinaed the whole thing again. When you look at Fulham's three home games, they will be thinking they have still got a squeak.
"It's still all to play for, but we've made huge strides - like we did against West Brom here. The ridiculous thing about the Premier League is that we've played better in the last two games by far and haven't picked up anything. We have done enough today to get the result.
"First half, there was a lot of nervousness and edginess around - which is only normal for a club like ours. We have got Boydy in particular that I'm delighted for because you know it's your day when he scores with a header!"
Asked whether a day out at Wembley next Sunday has caused nerves to start fraying, Bruce added: "It's the whole Premiership. We have got a lot of players who are new to it.
"Boydy is the typical one - six years ago he was playing non-league. David Meyler hasn't really played in the Premier League, Jake Livermore hasn't really played. It's new and it's a new experience for them.
"They can only get better with it because it is a big, tough league. Their resilience and their attitude has put them where they are in the table today."
One man with plenty of experience, and one who has become a key figure for Hull, is Tom Huddlestone.
He was forced to sit things out against the Swans after taking a knock, but Bruce is confident the England international midfielder will be ready to face Sheffield United.
He said: "We expect him to be. He hurt his back last week, he was sore around his midriff, so we'll see how he is next week. We expect him to be fit."