Cheikhou Kouyate eased their relegation fears, and lifted the heat on manager Bilic, with his first Premier League goal of the season.
Five straight defeats had left West Ham in serious danger of being sucked into the battle at the bottom, and a sixth would have left them just two points ahead of Swansea, and possibly signalled the end for Bilic.
But instead they can breathe easier after moving eight points clear of the drop zone.
"Forget me, it's part of my job, but with the amount of pressure the players were under they gave everything," said Bilic.
"They are the ones performing in front of 60,000 people. I'm pleased with the way the boys did it, it wasn't easy for them.
"They are well paid and all that but they are in their 20s, the pressure was huge, so I was more than pleased.
"But we are the ones who put ourselves under this kind of pressure."
Kouyate scored the crucial goal, moment after Michail Antonio went off with a hamstring injury, just before half-time.
The Senegal midfielder embarked on a marauding run into Swansea's half, exchanged passes with Robert Snodgrass before taking aim from 20 yards and lashing the ball inside the near post.
Bilic was given another vote of confidence by David Sullivan before the match, and he got a pat on the back from his other chairman, David Gold, afterwards.
"Mr Gold was just in my office and he is delighted," added Bilic. "I speak to Mr Sullivan the day after every game.
"It was a good day for us, especially when it's against a team below us and opened up the gap. It's really good."
For Swansea, a fifth consecutive away defeat leaves them in 18th, although they will at least be heartened by Hull losing at Manchester City to keep them in what increasingly appears to be two-horse race to avoid the last relegation spot.
However, manager Paul Clement knows they have to improve to stand any chance of survival.
"It was a poor performance and not what we were looking for," he said.
"But it was a poor game, lacking in quality, two teams playing with a lot of anxiety. There was a lot of fear in our play and although it improved slightly in second half from a battling perspective, the technical quality was really disappointing.
"We can't make excuses, it wasn't good enough. When you are in a battle for survival you expect the basics to be better. We made a huge amount of mistakes.
"We have to deal with that anxiety or we will be playing in the Championship next season."