After eliminating Serena Williams on Tuesday, Kvitova took little time to make it through round three, seeing off her opponent 6-4 6-2.
A victor at the ATP Masters 1000 event in OH as recently as 2015, Federer was playing for the first time since turning 37 last week.
In men's action, Andy Murray's comeback following hip surgery hit another roadblock on Monday after he was beaten 6-1, 1-6, 6-4 by France's Lucas Pouille in the first round of the Cincinnati Open.
Goffin avenged a quarter-final loss to world number 15 Tsitsipas in Washington two weeks ago and will next face Frenchman Benoit Paire.
Playing his fourth tournament after missing nearly a year due to an ongoing hip problem, Murray tallied eight double faults and was broken four times in a rusty display. She has dealt with blood clots and recently said she has been struggling with postpartum emotions.
"People that are sitting there during the week watching that all, I don't think you can plan to do that", he said.
"I'm still at the very beginning", said the 23-time Grand Slam champion.
"I competed too hard in the first set to sort of just let the match slip and not compete, give myself a chance to win, at least".
The second-seeded Swiss, has won in Cincinnati a record seven times but hasn't played there since winning in 2015.
Navjot Singh Sidhu arrives in Pakistan to attend Imran Khan's oath-taking ceremony
Khan is assured to win, following his party's victory in last month's general elections. Mr Khan is to be sworn in as Prime Minister on Saturday.
"This year has been tough".
But Gojowczyk proved unable to take them, Federer later coming to the net and putting away a confident backhand volley to hold to love and take the opener.
Spain's Pablo Carreno Busta, Briton Kyle Edmund and Canadian young gun Denis Shapovalov also advanced to the second round. After surviving five match points, the British No 1 eventually succumbed 6-4, 7-5.
"I've got an eye on that, but then I'm obviously going to re-evaluate and see what I can do for my body moving forward".
The Australian Open semi-finalist Hyeon Chung won the last five games to beat Jack Sock 2-6, 6-1, 6-2.
Tenth seed Novak Djokovic overcame home favourite Steve Johnson, needing nine match points to advance to the second round 6-4 7-6 (7/4).
Tenth-seeded Julia Goerges was losing to Kristina Mladenovic 6-4, 3-2 when she retired because of a lower left leg injury.
Peter Gojowczyk of Germany defeated Joao Sousa 2-6, 6-4, 7-5 and will next line up against seven-time victor Roger Federer, at second the highest seed in the field after the withdrawal of Rafael Nadal.