Bruce set his team the target of achieving 10 victories ahead of the season, and they reached that figure thanks to George Boyd’s 39th-minute header.
It sent Hull nine points clear of third-bottom Fulham, who won 2-1 at Aston Villa, with five games remaining.
Hull travel to Wembley next weekend to face Sheffield United in the FA Cup semi-finals and Bruce is delighted to visit the national stadium with top-flight security all-but confirmed.
"We've not safe mathematically, but it was giant step towards it," Bruce told reporters. "I've said all along that we need 10 wins. You'd be very unlucky to win 10 and go out of division.
"But it's ridiculous. All of a sudden that result for Fulham and West Brom winning at Norwich has concertinaed the whole thing again. You look at Fulham's three home games, they'll be thinking they've got a squeak.
"It's still all to play for. We're nine points clear – you could say 10 with goal difference – but it's not over. We can't take anything for granted but that's given everybody a lift."
Bruce believes Hull refusing to play on the counter-attack, as many others do against Swansea, worked in their favour.
"We wanted to get right up against them," he explained. "We watched them against Norwich. Norwich sat off them and if you sit off Swansea, they pass it around you. So we decided to go up against them.
"It was difficult to do it the whole game but I thought we did it fantastically. To keep a clean sheet, again, was terrific.
"The players' resilience is there for all to see. The reason we are where we are is because we've got a great attitude and we're determined to see it through, and thankfully we've done that today.
"The ridiculous thing about the Premier League is that we played much better in our two previous games by far and picked up nothing. Today, well, we've done enough."