Seeded ninth, Begu loomed as a tricky first-round opponent for Sharapova, but the former world number one from Russia swept to a 6-4, 6-2 victory.
However, two-time Wimbledon champion Petra Kvitova suffered a shock defeat, going out to local hope Zhu Lin in three sets.
Currently ranked 86th after returning from a 15-month doping ban, Sharapova — granted a wildcard in Tianjin — plays unseeded Magda Linette of Poland next.
The 30-year-old Sharapova, who returned to tennis in April after serving a suspension for taking the banned substance meldonium, has yet to win a tournament since then.
The five-times Grand Slam winner exited the China Open last week in the third round at the hands of Romania s Simona Halep, the newly crowned world number one.
But Sharapova had no such trouble against Begu, seeing off the 57th-ranked Romanian in 82 minutes.
The opening matches of the Tianjin Open have been badly disrupted because of rain and Sharapova said she was glad to finally get into action.
“It s been a long wait for everyone, it hasn t been easy,” the Russian was quoted as saying by the official WTA website.
“I had a first practice outdoors and then I think it rained for 70 hours straight, unfortunately for the players.”