BOSTON -- For one day, everything was fine again with the Boston Red Sox.
"Hopefully it's the start of a long run," said Gonzalez, who finished with three RBIs.
Carl Crawford added three singles, an RBI and a leaping catch for the Red Sox on his 31st birthday.
The victory snapped Boston's four-game losing streak and salvaged a rough weekend against the AL's second-worst team.
Coming off consecutive late-inning losses -- one in extra innings and the other when they blew a two-run ninth-inning lead Saturday, the Red Sox rebounded behind a strong showing by Morales (3-2).
Grumbling about manager Bobby Valentine's decisions and the lack wins can take a break for at least a day. The underperforming Red Sox moved to within a game of .500 again (54-55).
The 26-year old Morales, taking Josh Beckett's turn in the rotation after the righty left his previous start with a back spasm, gave up one run on three hits in six innings.
Now, Valentine just needs to find a regular spot for him in the rotation.
"I think it's an easy decision. We just have to figure out all the parts," Valentine said. "He wants to start and I want him to start. We can't always count on a guy getting injured to get him out there."
Gonzalez also understands how well Morales has pitched in a spot role.
"Yeah, it's really impressive," he said. "Every time he's out there he does a good job.
Ben Revere extended his career-best hitting streak to 19 games with a single, drove in a run and made a splendid over-the-shoulder catch for Minnesota.
On Saturday, Joe Mauer's three-run homer with two outs in the ninth capped a comeback win for the Twins.
"We've had a pretty good feeling. One big hit, a couple of walks, take some pitches and try to make them throw some pitches," Minnesota manager Ron Gardenhire said. "You get a couple of guys on, you never know what's going to happen. We've had some success battling, we've been in a lot of close ballgames lately. We're not going to quit. I can guarantee you that."
Alfredo Aceves, the losing pitcher Saturday, got the final three outs for his 23rd save.
The Red Sox hadn't been swept in a four-game series at home by Minnesota since 1991.
Morales has given up two runs or fewer in five of his six starts this season.
Revere's hitting streak is the longest by a Twins' player since Torii Hunter had a 23-gamer in 2007.
Leading 2-1 with two outs in the fifth, Gonzalez hit a drive into seats above the Green Monster off starter Nick Blackburn (4-7) after Crawford singled.
Blackburn's road woes continued. He gave up four runs on nine hits in five innings to fall to 17-35 in career road starts.
Revere robbed Jarrod Saltalamacchia, running full speed to the center-field wall to make the grab on the track and then turning to throw to second to complete a double play.
Willingham and Doumit homered off reliever Vicente Padilla in the ninth.
Blackburn had a feeling the Twins were going to do it again.
"I thought we had a decent chance there in the ninth," he said. "It's exciting to see."
Boston had jumped in front 2-0 with a pair of runs in the third. Crawford had an run-scoring grounder after Mike Aviles singled and Ellsbury doubled. Gonzalez followed with an RBI single.
Minnesota cut it to 2-1 in the fifth on Revere's sacrifice fly. Alexi Casilla walked, stole second and advanced on Jeremy Carroll's ground out.
Crawford, who wore his uniform pants high with his bright red socks showing nearly to his knees, robbed Doumit with his second-inning catch at the wall.
The two clubs have played seven four-game sets in Fenway Park since the last time the Twins swept a four-game series.
During the game the Red Sox acquired Twins 3B Danny Valencia for Class A outfielder Jeremias Pineda. The 27-year old Valencia hit .198 with two homers and 17 RBIs in 34 games with Minnesota this season. He was assigned to Triple-A Pawtucket. Pineda is hitting .421 in 36 Gulf Coast League games. ... OF Willingham was back in the starting lineup after resting Saturday with a right-hander going for Boston. ... Valentine said "everything is on track" for Beckett to start Wednesday against Texas. ... Boston CF and leadoff hitter Ellsbury returned to the starting lineup after resting what Valentine called a minor leg injury Saturday. ... Red Sox righty Aaron Cook (2-5) faces Japanese star and Rangers right-hander Yu Darvish (11-7) at Fenway on Monday night.