Smith, ranked 156th, needed only 70 minutes to dispatch Tomic, taking 15 of 23 points off Tomic's second serve and surrendering only one service break to the world number 25.
Smith had been 0-5 in ATP matches after first-round losses this year to South African Kevin Anderson at Delray Beach and Frenchman Kenny De Schepper in five sets at Wimbledon.
Next up for Smith will be 166th-ranked American Jared Donaldson, who beat 125-rated compatriot Austin Krajicek 6-7 (5/7), 6-3, 6-2 in another opener on the grass courts outside the tennis legends honor hall where induction ceremonies will be staged Saturday.
Croatian second seed Ivo Karlovic blasted 11 aces and never faced a break point in battling past Ukraine's Illya Marchenko 7-6 (7/4), 6-7 (6/8), 6-4. Karlovic took the only break of the match to advance after two hours and 23 minutes.
Japan's Tatsuma Ito outlasted US wildcard Noah Rubin 6-3, 6-7 (7/9), 6-4 to book a second-round date with US seventh seed Steve Johnson, who downed Germany's Michael Berrer 6-7 (6/8), 6-1, 6-2.
Also moving on was Colombia's Alejandro Falla, who ousted Spain's Adrian Menendez-Maceiras 6-1, 7-5.
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