If blood counts, then the Timberwolves left it all on the floor Tuesday night in a 112-97 ­victory at Utah that started an important four-game road trip.

They thumped the Jazz for the second time in four nights, building a 26-point, third-quarter lead and winning comfortably before they flew off into the night to San Francisco to prepare for challenging games Friday at Golden State and Saturday at ­Portland.

Before they went, Wolves head athletic trainer Gregg Farnam and his rubber gloves deserved the night’s game ball for cleaning up two bloody messes that delayed play slightly while arena crews were called to clean up.

Reserve guard Alexey Shved played fewer than two minutes late in the first quarter when he was struck across the face by a Jazz player and left the game for good because of a broken nose. Starter Kevin Martin left late in the third quarter when he lacerated the pinkie finger on his shooting hand while reaching into Utah guard Alec Burks’ mouth searching for a steal. “I’ve never seen that much blood on the floor with two players,” Wolves guard Ricky Rubio said.

While workers wiped the floor of Martin’s blood, Rubio participated in a playful game of catch with an adorable little girl seated courtside on a night when he masterfully orchestrated a Wolves offense that recorded 31 assists on 41 made baskets, and four starters scored 18 or more points.

“She was having fun,” Rubio said about his game of catch, “and I was, too.”

It helped that the Wolves were about to build a 26-point lead after they had led the Jazz by 36 points at home on Saturday. Rubio delivered an 11-point, 13-assist, five-rebound, five-steal performance.

He said he felt “comfortable,” perhaps like his old self, while Kevin Love and Nikola Pekovic again asserted themselves physically against an overmatched Jazz team.

“When we share the basketball and we defend, we are a hard team to stop,” Rubio said.

“When things are not going well, everybody’s trying to go his own way and that’s something we have to correct and be a ­better team.”

The Wolves moved back within a game of .500 at 20-21 Tuesday, when Utah got scorer Gordon Hayward back but lost starting center Derrick Favors to the same sort of hip-flexor injury that sidelined Hayward for ­Sunday’s game and four others.

They also beat the Jazz in Salt Lake City for the first time since Jonny Flynn supplied the winner in December 2009.

Without Favors, the Jazz had one fewer option on defense on a night when Pekovic went for 18 points and nine rebounds, Love had a 19-point, 13-rebound night and everybody again had fun, except, that is, Shved and Martin.

“I told Lexey he’s got to learn how to draw a foul,” Wolves coach Rick Adelman said. “He really got hit, and Kevin [Martin], too.”

Shved pleaded for a foul call to official Kane Fitzgerald by going to midcourt to make his case at first quarter’s end and for emphasis, bled on the floor in front of him to show where and how hard he had been struck.

“Shved was running like a little chicken, he was bleeding all over the place,” Pekovic said. “For everything, there’s a first time.”

Martin avoided needing stitches in his finger when it suddenly stopped bleeding and a doctor determined his finger isn’t broken.