Joel Campbell put the hosts in front before Swansea responded through Wayne Routledge and Ashley Williams.
While the result will have big repercussions for the Premier League title race, it will also impact on the relegation battle as Swansea moved six points clear of 18th-placed Norwich City.
Curtis, who took charge of the team after head coach Francesco Guidolin was hospitalised with a chest infection, said the win was a significant step in the right direction for Swansea.
"That's a very important win for us," he told BBC Sport. "We had our backs to the wall but I thought we played really well.
"We had a bad start, deservedly went behind, but in the second-half I thought we were excellent.
"It's a great atmosphere in the dressing room and tonight was a great reminder of how it feels to win games at this level.
"Hopefully we can get a few more before the end of the season."