Price (4-4) allowed two runs on only five hits and walked none on 95 pitches. He came within one strike of his first complete-game shutout since June 12, 2015, with the Detroit Tigers, but Manny Machado spoiled the bid with a two-run homer.
Bogaerts hit a three-run shot, his second in as many days and his sixth homer this season, while Martinez knocked a two-run bomb. Martinez tied the major league lead for homers (13) and RBIs (38) with his blast.
Machado's home run was his 14th of the season for Baltimore. Orioles starter Kevin Gausman (3-3) allowed all six runs on eight hits with two walks and six strikeouts in a 4 2/3-inning defeat.
Boston won its second straight game after beating the Oakland Athletics 6-4 on Wednesday to avoid a three-game sweep. Baltimore had won five of seven coming in, including a 17-1 rout of the Tampa Bay Rays last Sunday.
Thursday's game was a makeup for the annual Patriots' Day game scheduled for the morning of Monday, April 16. The game was postponed due to rain, marking the Red Sox's first rainout of a Patriots' Day game since 1984.
Martinez's two-out homer sailed an estimated 431 feet into the center field bleachers only four batters into the Boston first following a leadoff single for Mookie Betts, who proceeded to steal second base.
Bogaerts' bomb (estimated to be 383 feet) cleared the Green Monster in left and chased Gausman from the game with two outs in the Red Sox's fifth. Andrew Benintendi plated Jackie Bradley Jr. on a sacrifice fly earlier in the inning.
Martinez left the game in the seventh with a stomach illness. Blake Swihart replaced him in left field.
Machado's homer over the Monster came with two outs in the Baltimore ninth.
Before the game, Baltimore recalled catcher Andrew Susac from Triple-A Norfolk and optioned catcher Caleb Joseph to Norfolk. Susac, who started and batted ninth, went 1-for-3 with a strikeout in his season debut.
Boston second baseman Dustin Pedroia (left knee surgery recovery) will rehab again with Triple-A Pawtucket on Friday. Red Sox manager Alex Cora said Pedroia could return to the team before the end of May.
Red Sox left-hander Drew Pomeranz (1-1, 5.47 ERA) opposes Orioles right-hander and Massachusetts native Alex Cobb (0-5, 7.06) on Friday night.
--Field Level Media