Moreland provided three hits, Pierzynski added two with two runs scored and each socked a solo shot in the fourth inning, enabling Texas to grab a 3-1 lead.
Elvis Andrus added a two-run single in the seventh to ice the victory for the Rangers, who improved to 5-2 to start the season in taking the opener of a three-game series.
Alexei Ogando (2-0) tamed the Rays on one run over 5 1/3 innings, yielding three hits and three walks while striking out a pair.
Joe Nathan recorded his second save despite allowing a run in the ninth.
Rays starter Jeremy Hellickson (0-1) absorbed the loss, allowing three runs on five hits over five full frames.