Ortiz hit a solo homer in the first inning and added a two-run shot in the third off Vicente Padilla (0-2) that put the Red Sox ahead to stay. It was his 27th career multihomer game.
Schilling (1-1) allowed only one run and four hits over seven innings with six strikeouts and a walk. He retired the last 10 batters he faced and the only Texas runner who got past first base against him was Frank Catalanotto, who hit a solo homer in the first inning.
In Boston's season opener at Kansas City, Schilling allowed five runs and eight hits in four innings. That was the first time since July 2001 that the 40-year-old right-hander didn't make it past four innings.
Jonathan Papelbon got the final five outs for his second save. The hard-throwing right-hander came on with two runners on in the eighth, getting Michael Young on a called third strike before Mark Teixeira's infield popout.
Joel Pineiro relieved Schilling to start the eighth and walked the first two batters before Kenny Lofton reached on an infield single when trying to advance the runners with a bunt. Pinch-hitter Nelson Cruz, the only batter faced by Javier Lopez, drove in a run with a fielder's choice grounder.
Padilla also allowed only four hits over seven innings, but gave up the homers to Ortiz. He struck out four, walked one and retired the last nine batters he faced.
After leading the AL with a team-record 54 homers and 137 RBIs last year, Ortiz started this season with an RBI double in his first at-bat. He didn't drive in another run until the finale of the team's season-opening six-game road trip.
The only Red Sox homer in the first five games was hit by Kevin Youkilis.
Boston finally plays at Fenway Park on Tuesday against Seattle.
Catalanotto was 0-for-10 in his return to Texas before he pulled a ball down the right-field line that was just inside the pole to tie the game. Young followed with a single, but the Rangers had only three more baserunners against Schilling.
Lofton drew a leadoff walk in the third but couldn't get back to the bag on Young's flyball and was retired on an inning-ending double play. The Rangers re-signed Padilla this winter, giving him a $33.75 million, three-year contract after he had a career-high 15 victories in his first season in Texas. He won his first two starts a year ago. Ortiz's first homer came on a 94 mph fastball thrown over the middle of the plate, a line shot that cleared the fence in right-center field. Alex Cora, who started at second base to become the last Boston player in a game this season, was hit by a pitch to start the third. Ortiz then hit his second homer to right. NOTES: Catalanotto, who hit .330 for Texas in 2001 and spent the last four seasons in Toronto, had never had a longer hitless streak to start a season. ... Rangers 1B Teixeira was presented his 2006 Gold Glove award, his second in a row. Former Rangers C Jim Sundberg, a six-time Gold Glove winner, took part in the pregame presentation.