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Blake, 32, who had failed to take advantage of his first six opportunities to reach the milestone this season, seemed to be heading for another disappointment as the 20-year-old Harrison dominated the opening set.
But Blake, now ranked 110th in the world, found his rhythm and sealed the victory against his 43rd-ranked opponent with a late break of serve in the third set.
Blake next faces Australian Matthew Ebden, who ended a three-match skid with a 5-7, 6-1, 6-1 victory over Ukrainian Sergei Bubka.
Eighth-seeded Go Soeda of Japan shrugged off a lengthy rain delay to defeat Belgian Xavier Malisse 4-6, 7-5, 6-4.
The match had been level at 2-2 in the opening set when rain stopped play for more than two hours.
Soeda, who reached his first career ATP semi-final at Chennai this season, rose to a career-high ranking of 54th in the world this week. He next faces Russian Igor Kunitsyn.
Belgian qualifier Ruben Bemelmans defeated American Rajeev Ram 6-1, 6-3 to line up a second-round clash with top-seeded American John Isner.
Isner, who on Sunday claimed the title at Newport for the second straight year, is using Atlanta as preparation for the London Olympics, which will hold the tennis competition at Wimbledon.
Qualifier Ricardas Berankis of Lithuania, playing his first ATP main draw of the season, also won his match against Dmitry Tursunov 6-3, 6-2.
He next faces third-seeded Kei Nishikori of Japan.
