Both players lost their serve three times in the first set but Boulter dominated the tie-break to take an advantage.
Boulter improved significantly in the second set, when she won 78% of points on serve compared to 51% during the first set.
The victory was much-needed after a disappointing summer.
The 29-year-old came into the match on the back of five opening-round defeats from seven tournaments, slipping behind Sonay Kartal in the rankings to become Britain’s number three.
Cirstea came from behind to beat Japan’s Moyuka Uchijima 2-6 6-4 6-2 to set up the match against Boulter.
Top seed Naomi Osaka strolled into the second round with a comprehensive 6-0 6-4 win against Sonobe Wakana.
AloJapan.com