The Red Sox have now won four straight games and improve to 81-61 on the season. The Rays, who will play a "home" series against the Yankees at Citi Field in NY next week to avoid the aftereffects of Hurricane Irma, are fighting to stay in the AL wild card race but dropped two games under.500 (70-72) with Friday's setback.
Danny Espinosa had three singles for the Rays. In his career, Andriese is 2-1 with a 4.11 ERA and 32 strikeouts against Boston.
Carson Smith, Heath Hembree and Blaine Boyer each worked an inning of relief.
Before his departure, Nunez collected a hit in both of his plate appearances, scoring the first run for Boston in the first inning.
Andriese made his first start since coming back from a hip injury at the Chicago White Sox on Sunday and did not fare well, surrendering five runs and six hits in five innings.
In the second, Benintendi had a two-run double off the Monster, lefty hitter Moreland a two-run single after Mookie Betts, a right-handed batter, was intentionally walked by righty Andriese, and the fifth run scored on third baseman Evan Longoria's throwing error.
BOSTON (AP) - Tampa Bay Rays outfielder Steven Souza Jr. left a game Friday night after colliding with the right-field wall at Fenway Park. He returned to the lineup after missing Friday's game with minor back spasms, then singled in his first at-bat, stole second but got up gingerly after a headfirst slide.
"I think we came out with good energy", Red Sox manager John Farrell said after Wednesday's game moved his team to 12-4 on the season against Toronto.
He is 2-11 with a 5.06 ERA lifetime against the Red Sox - 1-3 with a 6.23 ERA at Fenway - as he and his teammates try to keep the Rays in the hunt for an American League wild card spot. Lucas Duda added a homer in the seventh, his 29th of the season and 12th with the Rays. He's 1-2 in his last three with a 6.48 ERA.
Red Sox: RHP Rick Porcello (9-16, 4.67) is in line to start.