The 20-year-old Chelsea player, who is on a season long loan at Turf Moor, scored twice in the opening 23 minutes with magnificent left-footed strikes that gave home keeper Steven Benda no chance.
"That second goal just travelled and kept going. Ian has given so much defensively and then he was able to add that to his game - not many players can give you that," said an admiring Burnley manager.
"The set-up for the free-kick is never a coincidence, but then how he hits it he has to take the credit for that."
Maatsen struck first from a free-kick outside the Swansea box in the 12th minute and then added his second with a thunderbolt from the left edge of the box. Swansea replied with an Ollie Cooper goal in the 27th minute.
"We started the first half extremely well and recovered a lot of balls and created moment, and changes in transition and out of our possession," added Kompany.
"Then for about 20-25 minutes they were better than us and on top of us. That's because they changed it, as Swansea do. If you get a press on them and get a hold on them they change it around and move positions.
"We weren't able to get the same pressure on them. If Swansea have time on the ball then they are going to hurt you. We went a little bit more compact, but that didn't work massively for us. In the second half it was an even game.
"We got a press on for the full 45 minutes in the second half and that made us a better team. We also had the better chances in the second half, although the result is the most important thing.
"We burned up a lot of energy and I could see a lot of tired legs at half-time. Then in the second half they just went again. That's the strength of this team - they just keep going when you think they are about to bend or break."