The Hungarian made his first Premier League start at Arsenal and, despite letting in four goals, Coyle believes Bogdan could take plenty of pride from his display.
The Scot said: "I thought he performed ever so well. When you go to clubs of the stature of Arsenal, there's going to be times when you're under pressure.
"He made some great saves and he was unfortunate with a couple of the goals. I think if anything he'll have taken great confidence from the match and he goes to Aston Villa with our full backing because potentially he's a top, top goalkeeper."