Wenger's side had fallen six points behind leaders Chelsea after drawing three of their last four league games and they were in danger of another damaging setback at the Emirates Stadium on Sunday.
Having led through Alexis Sanchez's early goal, the Gunners were pegged back by Callum Wilson's controversial penalty midway through the first half.
With the Arsenal faithful getting restless, Wenger revealed he had tried to calm his players down at half-time as they wrestled with the anxieties caused by three successive draws.
They responded to Wenger's pep talk by scoring twice after half-time through Theo Walcott and Sanchez to move within three points of Chelsea.
"We started well but when they equalised we became nervous," Wenger said.
"That is where the three draws were on our mind. It was down to us to calm down at half-time and just focus on the game."
Wenger was frustrated by referee Mike Jones' decision to award Bournemouth's penalty for an innocuous collision between Wilson and Nacho Monreal and he was fearing the worst at that point.
"My view is we should have had a penalty on Sanchez and Bournemouth's was a very soft one," he said.
"It was shoulder to shoulder. I was really amazed he gave it.
"We became nervous and they became more dynamic. For me it's not exactly the side I wanted to play. They are a tricky side and keep the ball well."
Driven on by the irrepressible Sanchez, Arsenal eventually find their rhythm in the second half and Wenger was quick to pay tribute to the Chile forward, who has scored 10 goals this season.
"Sanchez, even though he looks dead he is still alive. He finds the resources to win the ball and do something special," Wenger said.
"He has the guts. He has that conditioning which allows him to always find an extra gear to be dangerous."
Wenger said he knows keeping Sanchez is essential if Arsenal are to challenge for major silverware, even if performances like this mean he could demand a bigger salary.
Asked if Arsenal were any closer to tying up a contract extension with Sanchez, Wenger added: "It is important, how important I don't know.
"We won't do that tonight. We have some time.
"You (the media) like us to spend money, so you should be happy!"
Bournemouth manager Eddie Howe was frustrated that his team left empty-handed despite an enterprising display and he took a shot at Jones for failing to award his team a second penalty for a handball by Monreal.
"We were well in that game, we created a number of chances. The 3-1 scoreline is difficult to take," he said.
"It's difficult here, especially when you are Bournemouth. A couple of things have gone against us.
"I'm not a fan of really soft penalties, but a bit of consistency seems to have gone out of the game.
"But we were without a number of players today and there wasn't much between the teams.
"The future of the team is good. I back us to be okay this season."