Walcott's goal, his 20th of the campaign, was the quickest by an Arsenal player in the history of the Premier League and the fastest by anyone in the league this season.
"We scored early and started well," Wenger said.
"He (Walcott) is 24 now. He is a mature player. He has improved his finishing. He has improved his ability to resist pressure from the opponents. It was as similar goal to the one he scored on Sunday against Manchester United.
"After that I felt QPR played with quality, pride and they were fearless.
"You can see that if they had not had to play with fear in the last few months they had the quality to stay in the Premier League.
"We needed a special focus today to win the game. The points were vital for us today. We couldn't make any mistakes and that was a bit detrimental maybe to our offensive game.
"They made life difficult for us. We had good focus and good determination. We lacked a few final balls in the final third.
"Overall, I expected a difficult game and I was not disappointed."