Luka Doncic scored 33 points and Kyrie Irving scored 30 points to lead Dallas to victory.
The second round will be held at the same venue on the 25th.
Doncic's last-minute performance, which scored 15 points in the fourth quarter alone, shone.
Dallas, which was pulled to 98-102 with 3:14 left in the final quarter, saved the chase with Doncic's 3-pointer.
With 1:56 remaining, P.J. Washington's 3-pointer turned the tide at 104-102, Dallas went on a 106-102 run with Doncic's jump shot again.
Seven seconds before the end, he was kicked out by a two-point margin of 106-104, but Irving and Mike Conley added points with free throws to seal the victory.
Minnesota's main player Anthony Edwards scored only 19 points due to Dallas' defense, and Karl-Anthony Towns also suffered a slump, scoring 16 points in the second half alone.