Swansea came into the match without a win in six games and fell behind at Villa Park to a Jordan Ayew strike.
But Sigurdsson's 25-yard free-kick got the visitors back on terms before Ayew's brother, Andre, nicked a late winner for Swansea.
"It was an important win, especially after going 1-0 down," said Iceland midfielder Sigurdsson.
"So we showed good character as a team to get back into the game and probably steal it at the end.
"We got back into the game with that free-kick and then changed the formation a little bit with Mo [Modou Barrow] coming on, it gave us a lot of pace down the wing.
"We kept going, knew we were going to get the chances.
"It was fantastic for the team because it's been a hard few weeks for us, having not won many games and lost a few points."
Bottom-of-the-table Villa had been on an even worse run and the loss to Swansea prompted Tim Sherwood's sacking as manager on Sunday.
But with a win to lift Swansea out of their mini-slump, Sigurdsson says morale will now improve and they can take added confidence into next Saturday's home game with Arsenal.
"We put the pressure on ourselves, we set targets before the season where we want to finish," he added.
"Of course we're not happy with the way we've been performing over the last couple of months.
"Sometimes you don't perform well, but you nick those wins, that's what it's about - as long as you're winning games you're happy.
"But of course we're not happy and we put the pressure on ourselves to win those games, and there has been some performances where we haven't been near our best.
"The only thing we can do is train hard and it will come.
"Days like these [against Villa] will help you with confidence and hopefully now we can kick on."
Gylfi Sigurdsson has praised his Swansea City team-mates for their come-from-behind win against Aston Villa.