Jefferson finished with 40 points and helped the Jazz build a 97-90 lead with 4:49 remaining when Barea was at the center of a critical play.
After Jefferson missed a jumper, Minnesota's Dante Cunningham rebounded. As Utah's Derrick Favors headed up the floor, he bumped into Barea and pushed him to the floor.
Favors was charged with a flagrant foul and Barea made two free throws.
When Ricky Rubio followed with consecutive 3-point shots he'd missed 29 of his previous 33 field-goal attempts Minnesota took a 98-97 lead and positioned itself to spoil Utah's playoff hopes.
"When [Favors] turned around, he probably didn't see and then he just ran me over," Barea said. "I don't think he did it on purpose, but it happens sometimes in basketball."
Did Barea think Favors should have been ejected?
"I don't know," he said. "I've got to see it. But it wasn't that bad."
Barea agreed that the game nearly turned on the play.
"It helped us get back in it," he said. "But they did a good job finishing the game."
In the first half, Barea kept the Timberwolves alive by scoring 19 points in just over 11 minutes. He scored six points late in the first quarter and 13 more during an eight-minute stretch of the second quarter.
"It gave us a lot of energy," Barea said. "I got in a good rhythm pretty quick. … But they are a good team. They made adjustments in the second half and started trapping me. It was a little harder for me in the second half."
Barea scored four points on 1-for-7 shooting in the second half, though the incident with Favors helped keep Minnesota close until the final moments.
"We're going to fight," Barea said. "We're going to fight every game. Coming to Utah, it's a tough place to play when they need a game. The fans are behind them, but it was a good test for us. It was the kind of game we're going to have to win in the future."
Said coach Rick Adelman: "Our guys competed tonight. Al Jefferson really got it going and, when we doubled him, they made threes. They did what they had to do."