The team's manager missed the deadline day for teams to sign new players because he was meeting the pontiff in Rome ahead of a football match for peace.
Arsene Wenger is a Catholic and said: "Meeting the Pope was an experience I did not want to miss."
He added: "I am a Catholic, so it was an experience, and one (invitation) I accepted a long time ago, and on top of that it was a game for peace and multi-religion understanding.
"I thought today where we are a bit in front of an international religious war (in Gaza), it was a very important game."
It was a surprise when former Manchester United player Welbeck signed for Arsenal and Wenger said: "If I had not travelled that day, Welbeck would not be here.
"I'll explain that a bit later, but the coincidence made that, because I was on my way. If I stayed at home, Welbeck would not be here today. I will tell you the story one day, but that's the truth."
He added: "He [the Pope] is a supporter of San Lorenzo in Argentina. He is a football fan, you cannot be born in Argentina and not be a football fan.
"He spoke with many Argentinians, so I wanted to be discreet and say 'it was nice to meet you and bye-bye'."