NEW YORK—His strokes awry, his emotions laid bare for all to see, Roger Federer figured out a way to stay in the U.S. Open.

Federer found himself locked in a five-set struggle against the sort of player the world is accustomed to seeing him dismiss with ease, and it was only down the stretch that the four-time defending champion at Flushing Meadows looked the part.

Avoiding as big an upset as tennis has seen in a while, Federer came back to beat 23rd-seeded Igor Andreev of Russia 6-7 (5), 7-6 (5), 6-3, 3-6, 6-3 Tuesday night to reach the quarterfinals at the year’s last...

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