Dallas converted five of its first six possessions, then jumped to a quick 11-6 lead.
Utah point guard Mo Williams picked up his second foul with 7:50 left in the opening quarter and was forced to the bench.
Marvin Williams scored 10 points in less than five minutes and Millsap grabbed nine first-quarter rebounds as Utah closed with one, 26-25, at the end of the first quarter.
Mo Williams returned to the court early in the second quarter and scored 10 points during a 15-8 run that gave the Jazz a 42-35 lead. Utah eventually led, 44-36.
Dallas fought back on the strength of 10-for-16 3-point shooting, however. The Mavs made six 3-pointers in the final 5:17 of the half on their way their eight-point halftime lead.
Darren Collison scored 13 points and O.J. Mayo added 12 to lead Dallas.
In the third quarter, the Jazz needed less than four minutes to tie the score, 67-67.
It was still tied, 74-74, when Mo Williams hit consecutive 3-point shots and a layup. Marvin Williams followed with a 3, and his basket with 1:41 left in the period gave Utah an 89-76 lead.