Mannarino took the first set and, although Federer cruised through the second, the Frenchman made him work hard again in the third before the Swiss completed the win.
Federer’s semi-final opponent on Saturday will be Belgian third seed David Goffin who beat American Jack Sock 7-6(6) 6-3 in the last of the night’s quarter-finals.
World number four Marin Cilic had an even harder time than Federer, needing two tiebreaks to beat 100th-ranked Hungarian Marton Fucsovics 7-6(3) 5-7 7-6(4) in just over three hours.
The Croat now faces Argentine Juan Martin del Potro, a late contender to snatch one of the last two places at the ATP World Tour Finals in London, who beat Roberto Bautista 6-2 2-6 6-4.
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