Federer and Khachanov traded breaks in the first two games but Federer made amends by grabbing a second one to restore his advantage and clinch the first set on his second set point.
Fourth seed Cilic battled past Gilles Muller of Luxembourg 6-3 5-7 6-4 following an hour's delay to the original order of play, while Lopez came through 7-5 3-6 6-2 after an interrupted encounter with Grigor Dimitrov later on Centre Court. Having watched Rafael Nadal compile La Décima three times on clay this summer, Federer will strain every sinew to post just one on the grass of Halle.
The 35-year-old defeated the Swiss master in the semi-final of Halle Open in 2016.
Federer, who has yet to drop a set in Halle after crashing out of Stuttgart in his first match back last week, limited his serve-and-volley game against the German who is a fine returner and Mayer was broken quickly as his opponent won the first set.
Zverev comfortably negotiated his first two rounds, winning both in straight sets, first beating Paolo Lorenzi in the opening round.
"I think the second part of the season will be really crucial for Andy, and Novak [Djokovic] as well".
Federer, will be seeded ahead of his Spanish rival for Wimbledon, which starts in eight days.
Italian Open champion Zverev had won all three of his previous finals this season, but was blown away by a vintage performance from the 18-time major champion in only 53 minutes.
His huge serve was on full display when he won the 's-Hertogenbosch title on grass last week and once again he used it to devastating effect against Querrey.
"It was important to be aggressive off the baseline and make him feel my variation and the power I can bring to the court", Federer said after the game.
"It's tough to play against a friend like Lucie", Kvitova said. "So I'm just going to enjoy it as much as I can". "For the others, I think Andy (Murray) was just extremely exhausted after getting to World number 1 at the end of a year ago".