T&T had posted 181 for three - the highest total of the CLT20 qualifying stage - with Darren Bravo (54) and Denesh Ramdin (50) both scoring unbeaten half centuries.
Uva Next's reply lasted just one ball as Shannon Gabriel dismissed Dilshan Munaweera, who was caught at third man for a golden duck, when the rain came and forced the match to be abandoned.
William Perkins opened the Trinidad innings with a breezy 15-ball 32 and Lendl Simmons scored 34 before being bowled by Umar Gul.
Evin Lewis' departure for one brought Bravo and Ramdin together with T&T on 86 for three and they built an unbroken stand of 95 runs from 62 balls.
The match was a dead rubber, with Yorkshire having qualified from Pool 2 into the main stages of the Champions League.