The result sees the Clarets move off the bottom of the Premier League table. Jay Rodríguez, Jacob Bruun Larsen, Zeki Amdouni, Luca Koleosho and Josh Brownhill struck for Burnley. Oli McBurnie was sent off before halftime.
Kompany said: "It felt like our performances have been good for a while now. After the disappointment comes a moment of reward.
"I have to congratulate the lads. I'm happy for them.
"They were just raring to go. We were extremely hurt by what happened last week. The lads, I could barely contain them, they were ready to go and do something at Turf Moor.
"The lads deserve all the credit they can get it. It's not been a one-off. We've had performances like that this season."