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.
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.
