The American, who won at Queen's in 2010, served 31 aces and saved all four break points he faced to beat his French opponent 6-7 (4), 7-6 (4), 7-6 (5).
Australian Bernard Tomic also came through a tough challenge, losing serve against American Tim Smyczek to fall behind 5-4 in the third set before sweeping the final three games to win 6-4, 3-6, 7-5.
"Very rarely you're going to get out of a match like that," Tomic said. "I'm happy I won ... realistically I should have lost."
Australian qualifier James Duckworth recovered from a disastrous opening set to beat Israel's Dudi Sela 0-6, 7-6 (5), 6-4 and set up a second round clash with Tomas Berdych, while Paul-Henri Mathieu of France earned a meeting with Andy Murray by beating Slovenia's Aljaz Bedene 7-6 (2), 6-4.
The tournament, a major warm-up for Wimbledon, has been at the Queen's Club for more than a century.
Murray, who won the title last year before going on to become the first British man to win Wimbledon in 77 years, is seeded three and bidding to join Lleyton Hewitt, Andy Roddick, John McEnroe and Boris Becker as a four-time winner at Queen's.
The tournament, led by top-seeded Stanislas Wawrinka and second-seeded Tomas Berdych, includes four former champions in Murray, Hewitt, Querrey and Marin Cilic.
The top eight seeds receive a bye into the second round.
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- Bernard Tomic
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- Tomas Berdych