The Swiss superstar received little resistance en route to the historic win, needing only 52 minutes to post a 6-3, 6-1 victory over lucky loser Yuichi Sugita of Japan.
"I appreciate these numbers way more today than ever before," said Federer. "I think I can embrace them more. It's a big number and I'm very happy to have a chance to hopefully add some more wins to that number."
An eight-time winner of this event, the 35-year-old Federer trails only Jimmy Connors (1,256) on the all-time wins list.
Federer was stunned by veteran Tommy Haas in the second round at Stuttgart last week, but he thoroughly dominated Sugita. Federer won 90 percent of his first-serve points and delivered eight aces to zero for his opponent.
Federer faced one break point in the match, did not commit a double fault and won the opening set in only 21 minutes.
"I played some good tennis, some nice points," Federer said. "I played the way I wanted to play: aggressive, took charge from the baseline and served well. Overall, I'm very happy."
Initially scheduled to face Yen-Hsun Lu, who had to withdraw, Federer boosted his record on the season to 20-2. His opponent in the second round will be Germany's Mischa Zverev, 6-4, 6-4 winner over Slovakian qualifier Lukas Lacko.
Third-seeded Kei Nishikori of Japan was among three of the top players in the field who needed three sets to advance, outlasting Spain's Fernando Verdasco 6-7 (7), 6-3, 6-4.
Fourth-seeded Alexander Zverev, the younger brother of Mischa, had an easier time with a 6-3, 6-2 romp over Italy's Paolo Lorenzi. Zverev will face countryman Philipp Kohlschreiber in the second round.