Harrison (6-6) snapped a two-decision losing streak, giving up one run on seven hits and a walk while striking out four.
Beltre hit a two-run homer to cap a three-run first inning and added a pair of doubles for the Rangers, who have won five of their last seven games. Michael Young and Josh Hamilton also hit two-run homers in the victory.
"I think that right now we are a better team than we were a few weeks ago," said Young. "We're a better team then we were a month ago and if we can keep that up we'll be in the hunt."
Ruben Tejada hit an RBI single in the fifth inning for New York, which started a six-game road trip against the AL with its fifth loss in eight games.
Before the game started, Mavericks forward Dirk Nowitzki took the mound to throw out the ceremonial first pitch 12 days after leading Dallas to its first-ever NBA title and winning the Finals MVP award.
Nowitzki, sporting a Rangers hat and jersey, and still wearing a small splint on his injured left middle finger, worked from the stretch and gunned a pitch that nearly sailed over the catcher.
He pulled his right sleeve up and flexed his biceps as he walked off the mound, waving to the receptive crowd.
It was, perhaps, a bit of good luck for the Rangers.
They scored three runs in the first inning, sparked by Ian Kinsler's leadoff double. Hamilton hit an RBI single in the inning and Beltre followed with his 13th homer of the season, snapping an 0-for-10 slump at the plate with a blast to left-center field to make it 3-0.
That was all until the fourth, when Elvis Andrus smacked an RBI base hit to score Yorvit Torrealba. Andrus was then injured on a head-first slide while stealing second and came out of the game with a reported left wrist sprain.
Acosta also surrendered Hamilton's homer in the eighth inning before he was yanked for D.J. Carrasco.
Pelfrey gave up four runs on eight hits in his six innings, while Acosta yielded the same number of runs on five hits on 1 2/3 frames.
"They swung the bat well today and their pitcher was throwing good," said Mets center fielder Carlos Beltran. "They had everything going on their side today and we will regroup and be ready for tomorrow."