The Premier League champions had to come from behind to see off the newly-promoted side at the Etihad Stadium.
"A lot of pressure was on us," Kompany told BBC Sport. "I thought we were the only team that could lose something today.
"We missed a lot of chances and Southampton played well, but we are happy with the three points. We have a few things we can improve on, but we always do that."
Next up for City is a trip to Liverpool on Sunday.