The skipper has hailed his side's fine recent form throughout the busy winter schedule and praised the "great game" enjoyed by Joe Hart against Tony Pulis's outfit
City saw off a determined Palace outfit due to a second-half goal from Edin Dzeko to ensure their perfect home record in the league remained intact.
"It's a massive win," Kompany told reporters. "Most of the players were as happy with this win as they were with the one against Liverpool."
"It was a tricky period, two games in 48 hours. But it's also six points in 48 hours, which is a great boost.
"I wouldn't say we won ugly. I just think we ground it out against a team that was really well organised. Credit to Palace, I was very impressed with the way they set up against us."
Kompany also made sure to praise goalkeeper Joe Hart, who has recently come back into the side after a spell on the bench and was one of his side’s standout performers on Sunday.
"I think we did the right thing by not giving them any chance to come into the game too much, even though Joe Hart had to make a couple of very good saves," Kompany continued.
"It's really hard for me. I'm a bit of a neutral on [the goalkeeping issue]. I was always very happy with Joe's performances but was also happy with Costel Pantilimon when he came in.
"We have got two great keepers, but it was undeniable that Joe pulled off a great game. But sometimes, it’s not the amount of saves you make, it's about doing the right save at the right moment, and that was the case with him."
"We've put ourselves in a nice position for the second half of the season. It feels like we've played so many games already, but we arrive in January and there’s actually still not much between all the teams."
Meanwhile, Manchester City manager Manuel Pellegrini claimed he was never doubtful over the result of Sunday's game, and talked up the merits of grinding out such narrow wins despite having become used to far more comprehensive scorelines at the Etihad.
"It's useful to win this way. If you ask me, I'd prefer to win by three or four but, for the team, it's useful to win this way.
"I talked with the players at half-time and told them to be patient and continue to try and find the space. I knew it would be very difficult because teams like Palace are prepared to defend and they do it very well. But I also had a lot of trust that we would score."