Of his decision to compete in Rotterdam Federer said: "You have to do it, not because it's the right thing to do, but you have to be here with all your heart and really go for it. Try my very best".
Andre Agassi, who now holds this record aged at 33 years and 133 days old. If the 20-time Grand Slam champion wins, he will take up the number one position for the first time since 2012.
Bemelmans was broken in his opening service game of the second set and despite offering some resistance was unable to threaten a dominant Federer, who claimed victory in the pair's first meeting in 46 minutes.
The Swiss showed no signs of nerves as he routinely thumped Bemelmans, losing just four points on serve and breaking four times.
'I did well there, I felt good, I'm healthy and I'm confident. "The beginning will be crucial tomorrow, you don't want to fall behind, you want to stay focused all the way".
The seventh-ranked Goffin, who beat Nicolas Mahut in straight sets in his previous match, will next face sixth-seeded Tomas Berdych.
But Zverev, playing in the last match on court, was downed by Andreas Seppi 6-4 6-3.
Goffin made light work of veteran Spaniard Feliciano Lopez, cruising through 6-1, 6-3 to reach the quarter-finals in his second match of the week.
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.
In first-round play, No. 2 seeded Grigor Dimitrov of Bulgaria beat Japan's Yuichi Sugita 6-4, 7-6 (5), while Andrey Rublev of Russian Federation ousted No. 7 seed Lucas Pouille of France 7-5, 6-4.
Dimitrov then sent a forehand into the corner to earn a pair of match points and advanced on his second with a service victor to end the resistance of the 41st-ranked Sugita.
None of the ties went the distance and, indeed, the first one up ended after four games, with Richard Gasquet retiring hurt when 3-1 down against French compatriot Pierre-Hugues Herbert.