The Gunners were in need of a response following a nightmare start to the season, which saw them sell both captain Cesc Fabregas and playmaker Samir Nasri before being thumped 8-2 at Manchester United ahead of the international break.
There was more than a touch of good fortune about their first win, secured by a well-taken effort from Andrey Arshavin following an error by Swans keeper Michel Vorm just before half-time.
For long periods the Welsh side more than matched Arsenal for enterprise and determination.
Wenger had included all five deadline day signings in the 18-man squad and was glad to get a first victory on the board.
"It was vital to win the game," said the Gunners boss.
"We started quite well, but slowly nerves took over and we played with the handbrake on in the second half.
"We just wanted to get over the line and did not make a mistake, but did not manage to score the second goal.
"Swansea keep the ball well and make it difficult, are quick on the flanks and have the basics of a very good Premier League team, they just do not score."
Despite a turbulent few weeks - which also saw Arsenal secure qualification for the group stages of the Champions League - Wenger is in no doubt things will soon turn around.
"Confidence goes quickly and comes back slowly. We are not used to the situation we have been in," he said.
"We will come back - we did not have a lot of players today and when everybody is back, we have a chance.
"But first, we need to put a few wins together."
Swansea had created the best chance early in the game when Danny Graham's close-range effort was turned behind by an acrobatic save from Wojciech Szczesny.
Theo Walcott saw his goalbound effort hacked off the line by defender Steven Caulker, on loan from Tottenham, before Vorm's error proved decisive.
His quick roll-out hit defender Angel Rangel before the ball dropped to Arshavin who slotted home from a tight angle.
A free-kick from lively Swans winger Scott Sinclair glanced the top of the Arsenal crossbar, while at the other end Robin van Persie struck the post and substitute Marouane Chamakh also headed a late chance at the keeper.
To their credit, Swansea refused to lie down, and almost snatched a point in stoppage time when Graham turned a loose ball over from a scrambled corner.
They were coached by assistants Colin Pascoe and Alan Curtis as manager Brendan Rodgers missed the game following the death of his father.
First-team coach Pascoe maintained Rodgers could be proud of his team's efforts.
"We have been in contact all the time, he only went last night and was there to see his father pass away," Pascoe said.
"I have spoken to him and he is proud of the way the lads played and performed.
"Everything had been prepared well for the game and were unlucky not to get a draw."
Pascoe added: "When you come to place like Emirates have to stop the flow of play, and we did that.
"We are gutted really that we did not get anything, but we have to score goals to win games. At the moment we haven't been doing that but we are creating chances, I would be worried if we were not."
Wenger paid Arsenal's condolences to his opposite number.
The Gunners boss said: "The team did fight for him and that is the best they can do for him."