Kontaveit had failed to win a tournament since her maiden WTA Tour triumph in 's-Hertogenbosch back in 2017 before defeating Begu 7-6 (7-5) 6-4.
The world No.30 from Estonia had lost two finals this year, but she was not to be denied in Cleveland.
Begu fought back from 5-2 down in the first set to force a tie-break, but one mini-break was enough for No.2 seed Kontaveit to take a big stride towards an elusive title.
The composed Kontaveit did not face a break point in a tight second set, not allowing her Romanian opponent a look-in as she served superbly.
Kontaveit only had one break-point opportunity of her own, but grasped it to lead 2-1 and went on to seal a confidence-boosting triumph ahead of the US Open.
Australia's Sam Stosur will be Kontaveit's opponent in the first round at Flushing Meadows, while Begu faces Andrea Petkovic.