Alexis Sanchez's seventh goal in his last six games had put Arsenal on course for a Premier League victory at the Liberty Stadium on Sunday.
Arsene Wenger's side had squandered a three-goal lead to draw 3-3 with Anderlecht in the UEFA Champions League on Tuesday and they failed to hold on to an advantage once again in south Wales.
Gylfi Sigurdsson equalised with a sublime free-kick 15 minutes from time, then Bafetimbi Gomis came off the bench to score his first Premier League goal with a towering header three minutes later to stun the visitors.
Victory for Swansea was only their second in eight Premier League matches, but it was enough to move them above Arsenal into fifth spot, much to the delight of manager Monk.
He told Sky Sports: "It's about belief. We talked about belief before the game, about how good we have been and how good we can be.
"We showed our character in the squad and it shows you how together we are.
"They got the goal, I thought it was an even game, we made the changes and thankfully they went for us today."
Monk hailed Sigurdsson's stunning strike, which sparked a dramatic turnaround and ensured Arsenal suffered only their second top-flight defeat of the campaign.
He added: "Gylfi Sigurdsson has been disappointed previously that he has not had many opportunities to produce what he did today but what a great goal and that put us in the ascendency.
