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The 23-year-old left boyhood club Manchester United in a £16m switch to the Gunners while Wenger was away in Rome for a charity game.
Wenger said: "If I'd stayed at home he wouldn't be here now."
He refused to elaborate on that but added he initially wanted Welbeck on loan with an option to buy.
However, Wenger said he was "happy" to make the transfer permanent - though he would not reveal whether  Welbeck will start against Manchester City on Saturday.
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"I cannot tell you if Welbeck will start but it's a possibility. I have to assess the squad," said the Frenchman.
Welbeck, who scored both of England's goals in the 2-0 win over Switzerland on Monday, is confident he can score plenty of goals for Arsenal if he is given a run of games as an out-and out striker.
"Once I get into the box and get the opportunities I have got faith in my ability," he said.
"Before I wasn't getting into the box because I wasn't playing too much as a striker and I had to worry about my defensive work. Once I get the opportunity and a run of games up front I will do my best."
Midfielder Aaron Ramsey could also be fit to start against the Premier League champions at the Emirates Stadium.
The player went over on his ankle in Wales' 2-1 victory over Andorra in midweek, which was played on a 3G pitch.
"His ankle injury will be assessed today, he has a little chance of playing on Saturday," said Wenger.
He said he was "relieved" the injury was not a bad one, adding: "I was haunted in my head about last year's injury, it was a little muscular injury and he was then out for three months."
Ramsey had earlier said he believed he would be fit for Saturday's game.
Arsenal manager Arsene Wenger says he would not have signed striker Danny Welbeck on transfer deadline day had he been in the country.