Strugglers Hamburg had battled hard to protect a goalless draw, hoping to end a three-game losing run, with goalkeeper Rene Adler making a string of saves before Kimmich drilled in a Franck Ribery cutback at the far post.
The result lifted Bayern to 15 points, three ahead of second-placed Borussia Dortmund, who beat Freiburg 3-1 on Friday.
Bayern, who take on Atletico Madrid in the Champions League next week, faced a determined Hamburg team, looking to turn their season start around, pressing them high and allowing just three shots on target in the first half.
Their first big chance came after almost an hour with Javi Martinez's effort cleared on the line by Adler, who was also on hand to save a curled David Alaba free kick 10 minutes later.
Bayern coach Carlo Ancelotti brought on Ribery and the Frenchman added instant pace down the left wing but neither Thomas Mueller nor Robert Lewandowski could beat Adler from close range.
The keeper was finally beaten when Ribery launched another trademark run, cut across the goalmouth and Kimmich slipped in to score the winner to extend Bayern's perfect start to the season with their fifth win in five matches.
Borussia Moenchengladbach climbed to fourth on 10 points after easing past Ingolstadt 2-0 for their tenth straight home win under coach Andre Schubert. Cologne, in third place on 10 as well, take on RB Leipzig on Sunday.
Hertha Berlin conceded a stoppage time equaliser from Michael Hector to draw 3-3 at Eintracht Frankfurt and drop to sixth on 10.
Mexican striker Javier Hernandez scored a hat-trick, including a stoppage time winner, to lead Bayer Leverkusen from a goal down and 2-1 down to a last-gasp 3-2 win over Mainz 05.
(Reporting by Karolos Grohmann, editing by Ed Osmond/Rex Gowar)
