BALTIMORE -- Boston's Jackie Bradley Jr. left on paternity leave before Tuesday night's game against Baltimore.

Manager John Farrell has a suggestion for the name of Bradley's first born.

Mookie Betts homered three times -- to center, left and right -- and drove in five runs to back the pitching of Eduardo Rodriguez as the Red Sox defeated the Orioles 6-2 for their third consecutive win.

"Jackie leaves us today because of the pending birth of his child," Farrell said. "If he has a boy, he ought to name him Mookie after tonight's outing."

It was Betts' first career three-homer game, and his five RBIs were also a personal best as Boston increased its American League East lead to three games.

Betts became the first Red Sox leadoff hitter to homer three times in a game in at least 100 years.

"He's probably one of the few guys I can say every time I see him hit I'm thinking he's going to hit a home run," Xander Bogaerts said. "It's just that presence he has in the batter's box."

Betts and Dustin Pedroia hit back-to-back homers to start the game against Kevin Gausman (0-3).

The Boston right fielder then lined a three-run homer in the second inning, and a launched a second solo shot -- this time off reliever Dylan Bundy -- in the seventh.

Betts, who has 12 home runs and 40 RBIs, lined out in the fourth and grounded out in the ninth.

"I was more concerned with just getting the fourth hit," he said of his final at-bat. "Home run would have been nice, but I was just trying to get another hit at that point."

He also made a diving catch in shallow right.

"For putting the whole game together I think today was (my) best day," Betts said, adding he hadn't hit three home runs in a game at any level of baseball.

Bogaerts extended his hitting streak to 24 games with a seventh-inning single.

Rodriguez (1-0) made his season debut after dislocating his right kneecap during spring training. He was solid, allowing two runs on six hits over six innings. Rodriguez, 23, struck out three without a walk.

"Threw a lot of strikes, he pitched comfortably," Farrell said. "Thought he had a very good changeup to go along with a fastball that I thought he spotted up well."

Rodriguez was aided by center fielder Chris Young, who robbed Manny Machado of a potential two-run homer with a leaping catch at the wall in the third inning.

"You go through periods where kind of everything falls your way, then you go through periods where they jump up and catch balls," said Orioles manager Buck Showalter, whose team has dropped seven of nine.

Machado and Adam Jones had two hits apiece for Baltimore.

Gausman went six innings, allowing five runs on five hits, but he retired 10 of the last 11 hitters he faced. He tied a career-high with 113 pitches while striking out eight and walking two.

Betts sent Gausman's second pitch of the night over the wall in center for his fourth career leadoff home run.

Pedroia homered to left center and Boston led 2-0.

In the second, Gausman walked No. 9 hitter Christian Vasquez to put runners on first and second with two outs and Betts lined a 2-0 fastball just inside the left-field foul pole to make it 5-0.

"Getting behind guys, not throwing quality pitches early in the game," Gausman said of Boston's early success. "Kind of seemed like the third inning, (I) kind of found it, and by then it was too late."

Baltimore got to Rodriguez in the third when Joey Rickard led off with a single and scored on a two-out double by Nolan Reimold.

Young limited the damage with his grab to rob Machado.

In the fifth, Baltimore got to within 5-2 when Jones singled home Paul Janish with two outs.

NOTES: The Red Sox have outscored opponents 54-16 in the first inning since April 11. ... Boston placed OF Jackie Bradley Jr. on paternity leave before the game and recalled OF Rusney Castillo from Triple-A Pawtucket. Castillo went 2-for-4 in one game in April for Boston. He is batting .241 with one home run and 13 RBIs for Pawtucket. ... The Orioles placed C Caleb Joseph (testicular injury) on the 15-day disabled list. Joseph was hit with a foul tip in Monday's game and had a surgical procedure Monday night. ... Baltimore recalled C Francisco Pena from Triple-A Norfolk, where he was hitting .200 in 24 games. ... Boston RHP Joe Kelly (2-0, 6.30 ERA) opposes RHP Mike Wright (2-3, 5.05) on Wednesday night in the third game of the four-game series.