The Swans fell behind to a Theo Walcott brace after a little more than half an hour before Sigurdsson scored the first goal of the new Bob Bradley era to keep his side in the game.
Mesut Ozil's strike was then cancelled out by Borja Baston to keep the deficit at one, but despite Arsenal being forced to play the last 20 minutes with 10 men, the visitors could not rescue a point.
"I am a little bit disappointed, but it was certainly an exciting football game. We didn't give up and could have nicked something at the end but it wasn't to be," Sigurdsson told reporters.
"The new manager wanted us to play on the front foot and I think we did that but we gave them too much respect. After the first goal, we started to play better and we had a lot of chances and could have taken something but overall I think it was a fair result.
"The boys really like the new boss and our training sessions are very tough. I'm not worried about my fitness. I'm fit. We will get fitter if we aren't fit enough."
The defeat leaves Swansea in the relegation zone with just four points from their opening eight games.