KANSAS CITY -- To get out of Kauffman Stadium with a three-game sweep of the Royals and a four-game overall winning streak, the Red Sox needed to cobble together a strength-in-numbers bullpen effort that would at least hold Kansas City in check.

Mission accomplished. Manager Alex Cora’s many trips to the mound paid off handsomely as the Red Sox used seven pitchers en route to a 7-5 victory. In the opening two games of the series, Red Sox starters Eduardo Rodriguez and Chris Sale combined for 19 strikeouts and no walks. On Thursday, however, a much different victory script was required that didn’t include dominant starting pitching.

• Box score

Starter Ryan Weber lasted just 1 1/3 innings, but a parade of relievers kept escaping jams and allowed the Boston offense to do its thing. Mookie Betts had a two-run homer and Christian Vazquez delivered a two-run triple as Boston surged back from an early 2-0 deficit.

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“The bullpen did an outstanding job,” Cora said. “Some big outs there with traffic.”

The Red Sox looked to be in trouble when the Royals strung together four consecutive hits in the second off Weber. With two runs in and runners at second and third, Colten Brewer came on to strike out Whit Merrifield and retire Adalberto Mondesi on a fly to center.

The Red Sox promptly struck for four runs in the third after escaping that jam.

Then in the fourth, it was Marcus Walden to the rescue.

The Royals loaded the bases with one out before Walden got Merrifield on a fly to short right and struck out Mondesi.

It was that kind of slip-and-slide day for the Royals as the Boston relievers occasionally bent but never broke.

“It was a bullpen day where we had to cover a lot of innings and each guy came out and did his job,” said Heath Hembree, who had his own escape act in the eighth when he came on with a runner at third and nobody out. Hembree proceeded to strand that runner at third, preserving what was then a three-run lead.

“It’s huge,” Hembree said. “For the bullpen guys to be able to go out there and piece together a win, it keeps our momentum. It wasn’t always smooth, and we had to grind through some things. But we were able to get it done as a group.”

Betts hit his two-run homer off Danny Duffy to tie the game in the third, and the Red Sox took the lead for good on a two-run single by Rafael Devers.

“I’ve just been trying to grind and do what I can to help the team score some runs,” Betts said.

The Red Sox (33-29) have gone 27-16 since their 6-13 start.

“We just have to keep playing good baseball,” Cora said. “We’ve got to get better. When we feel we’re clicking and playing our brand of baseball, then we’ll look and see where we are [in the standings]. Trust the process and we should be there.”