Dmitry Tursunov of Russia was eliminated in straight sets in the second round.
"It was a good match," Bogomolov said. "He was playing much better by the end of the match. He received perfectly when I was serving for the match at 5-3 up and broke me. I was lucky when he double-faulted on second match point. I'm happy that I will play in the next round."
The 53rd-ranked Starace broke Tursunov two times in the first set and in the fifth game of the second. Tursunov saved three match points at 5-4 before earning his only break point of the match, but the Italian served two winners and closed the match with a backhand volley.
"He (Starace) served well today while I rarely hit (the ball) with the center of my racket," Tursunov said.
"I knew it was going to be a tricky match because Berrer is a very good indoor player," Tipsarevic said. "It's never nice to win this way but I feel that even (though) it was 6-3, I did a lot of running in the first set."
Tipsarevic will play Starace in the next round.
The 84th-ranked Devvarman broke in the 11th game of the first set, but the 24th-ranked Cilic, who is looking for his first title this season, won five consecutive games in the second set and broke twice in the decider to advance.
Bogomolov is making his debut at a second consecutive event in Russia after reaching the quarterfinals at the Kremlin Cup last week. He said this week that he plans to represent Russia in the Davis Cup.
Kohlschreiber won three games in a row and saved a match point as he fought back from 5-2 down to tie it 5-all. But serving at 6-5 down and 30-all, the unseeded German returned wide and double-faulted on match point.