The Clarets, fresh from their opening 1-0 win at Huddersfield Town, trailed to a Dan Potts goal on five minutes as the Hatters started the better.
Burnley struggled with the visitors’ physicality in the first half, but the introduction of new signing Manuel Benson at half-time changed the game, as Josh Brownhill levelled matters, with the Clarets dominating the second half without finding a winner.
And Kompany said: "I think you always want to win, but I saw a spirit, high energy during the entire game, and in the beginning where I thought we were a little bit, for 10 minutes, on the backfoot dealing with what Luton brought to the table, after that we matched them for everything, won the second balls, were aggressive, got the ball in numbers and the energy started going into the performance, and you could see the fans had a lift.
"The players performed better in the second half and there were a lot of positives to take out of the game.
"It wouldn't be the first time I've been wrong in my life, but I think it's a play-off team (Luton), the way they're set up, they know very well what they need to do, they're very good at what they do, and I was impressed with how we dealt with that side of the game.
"We managed to create chances in the second half, and I think we can create even more and be more clinical, but that won't be perfect in the beginning for sure.”
And on Benson, whose electrifying pace and direct running had the fans off their seats: "He only trained a couple of times with the team and there are so many details in modern football that you want to bring to your players before you put them on the pitch.
"In the end, like any manager, if the game calls for it, you put him on, and that's what we did.
"We've discussed what someone like him can bring, and it's exactly what you saw - we couldn't deal with one v one situations down the side, we had the spaces in the first half and the same spaces in the second, but we just couldn't threaten enough.
"Every now and then on the side of Dara (Costelloe), we got in a couple of times and crosses were coming in, but it needed it from both sides, and Benson came on and changed the game, we need to call it what it is.”
The worry is that, for all their possession – and Burnley had over 70% for a second-successive goal, they didn’t work Luton keeper Ethan Horvath enough: "First half we didn't create enough for the possession we had, second half yes, for sure.
"We can always create more.
"You can have possession or no possession, but it's hard to create much more than we did in the second half, and we didn't give anything away, we were controlling the game as well, we weren't dealing with torpedoes coming at our goal for 90 minutes, we had the game where we wanted it to be.”