The Swiss began his quest to return to the top spot in the ATP Rankings for the first time in over five years with a 6-1, 6-2 trimuph over Ruben Bemelmans of Belgium at the ABN AMRO World Tennis Tournament in Rotterdam.
"It's very surprising how good things went today", Federer said.
Federer arrived in the Dutch port city at the weekend knowing a run to the semi-finals will move him ahead of Rafael Nadal in the ATP standings.
Federer will face off on Thursday against long-time German opponent Philipp Kolschreiber, who has lost a dozen times to the victor of a record 20 grand slam titles. "The key is to start well, put him under pressure". Andre Agassi, who now holds this record aged at 33 years and 133 days old.
Federer was on his game from the beginning and did not let up, giving up only four points on his serve in the first set while breaking his overmatched opponent four times.
Next up for the top-seeded Swiss on Thursday is a 13th career showdown against Philipp Kohlschreiber.
Eddie Jones accuses World Rugby of undermining officials
England head coach Eddie Jones says the training session with the Georgia Rugby team was a valuable learning lesson for his team. Wales assistant coach Rob Howley said: "It was good to have dialogue with World Rugby ".
Federer, who already holds the record for the most weeks spent at number one (302), said he "would do anything" to make history as the oldest number one. "Any lapses can get you into trouble".
The results: Second round: Andreas Seppi bt Alexander Zverev 6-4, 6-3; Tomas Berdych bt Viktor Troicki 6-1, 6-2; David Goffin bt Feliciano Lopez 6-1, 6-3; Grigor Dimitrov bt Filip Krajinovic 7-6 (4), 7-5.
Second seed Dimitrov, who beat Belgian fourth seed Goffin in the London final last November, had to fight back in the second set to defeat Japan's Yuichi Sugita 6-4, 7-6 (7/5) in their first-round encounter.
The Bulgarian was joined as a victor by rival David Goffin, who lost to Dimitrov last November in the tile match of the ATP Finals in London.
The Russian was joined in the second round at the Ahoy Arena by Pierre-Hugues Herbert of France, who advanced after just 15 minutes when compatriot Richard Gasquet, the Montpellier finalist, retired with a groin injury trailing 3-1.