City had to show plenty of patience to overcome dogged resistance from new-boys Brighton at a packed Amex Stadium.
But after a goalless first 45 minutes, City's superior quality eventually told with Sergio Aguero breaking the deadlock on 70 minutes and an own goal from Lewis Dunk sealing the three points five minutes later.
"The season is going to be long and today was an improvement on a lot of things we did last season," said Kompany.
"We had a five-minute spell where we were under pressure. But we put our bodies on the line, got some important tackles in and then we got the first goal as we were patient.
"So I think there were a lot of positives for us to build on.
"We had to be patient. In the first half they defended well and were well organised and you could see they had a clear game-plan.
"But you can see we have quality in the team and in the second half it made a big difference."
Kompany's new defensive colleague Kyle Walker enjoyed a fine debut for City, providing a real threat going forward on the right flank.
And despite only having been at the Club a few weeks, the England international revealed he was already revelling in life as a City player.
"I am enjoying it with the lads and they have been so welcoming," he said. "It's a big change after spending eight or nine years at Spurs but I am enjoying it and I am playing with a smile on my face.
"We knew Brighton would sit back and defend but we needed the win on the opening day so we are all well happy."