Playing in the one of the few mid-afternoon matches, Bogomolov beat Clement in front of an unusually large crowd on a side court.
Heavy rain halted the only three matches that were played Tuesday. Despite bright sunshine, center court and one of the two side courts were still soggy Wednesday.
Officials were unable to get center court in playable condition despite spending most of the day trying.
Three of the matches scheduled for one of the side courts were postponed.
In other opening-round matches, Nicholas Mahut of France beat American Amer Delic 6-4, 6-4; third-seeded Sam Querrey, the highest seeded American left, ousted Ruben Ramirez Hidalgo of Spain, 6-3, 6-2, while sixth-seeded Philipp Petzschner of Germany defeated Horacio Zeballos of Argentina 6-4, 6-7 (4), 6-2 in match that started Tuesday.
Two-time defending champion Fabrice Santoro of France had his match against Italy's Flavio Cipolla halted by rain tied 3-3 in the first set.
With fans two and three rows deep standing behind one of the baselines, 180th-ranked Bogomolov won his first ATP tour match since beating Querrey in the second round of Miami in March 2006.
"I don't know how to prepare for that," Bogomolov said of the unexpected crowd. "I don't know why it didn't bother me like it did him."
Bogomolov, who was born in Moscow but now lives in Miami, broke Clement's serve in the ninth game of the final set to take control.
He had left wrist surgery in March and lost four Challenger events this year.
"Tennis is tough," Bogomolov said with a smile. "I come back and lose four Challenger events, go to Wimbledon and lose in the first round and come here and beat the 59th player in the world."
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