Jordany Valdespin contributed an RBI double during the uprising and knocked in four overall for the Mets, who broke a three-game losing streak and took the opener of a three-game series.
David Murphy chipped in with a two-run single while Kirk Nieuwenhuis went 3-for-4 with three runs scored for New York.
Chris Young (1-0), making his second start of the year, earned the victory, yielding a pair of runs on nine hits over 5 2/3 innings. The right-hander notched his first victory since undergoing shoulder surgery last year.
Andrew Cobb (2-3) was saddled with the loss, surrendering six runs (five earned) on seven hits over 6 2/3 innings for the Rays, who fell into a tie with the Baltimore Orioles for second-place in the AL East, one game behind the New York Yankees.
Hideki Matsui and Will Rhymes drove in runs for Tampa Bay.
Notable deaths of 2014 [PHOTOS]