He rediscovered his rhythm, recovered the deficit and then broke the Tunisian in the 11th game before serving out the match.
Federer next meets Belgian David Goffin, who was leading Sergiy Stakhovsky 4-6 7-5 2-0 when the Ukranian retired hurt after an awkward fall.
"That's life...some days are just not meant to be. I'll try to be back for Wimbledon," Stakhovsky tweeted.
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Federer, 34, will be keen to enjoy as much match practice as possible before Wimbledon after pulling out of the French Open last month with a back injury.
He leads 25-year-old fifth seed Goffin 4-0 in their head-to-head tally.
The Swiss, ranked three in the world, is also joined in the quarter-finals by home favourite and eighth seed Philipp Kohlschreiber, who downed lofty Croatian Ivo Karlovic 6-7(7) 6-4 7-5 despite being on the receiving end of 26 aces.
(Reporting by Clare Lovell; Editing by Ian Chadband)
