Tiffany Hayes finished with 20 points and Alex Bentley had 18 in reserve roles for Atlanta, which was swept in the best-of-five finals for the third time in four years.
Moore was playing at the Gwinnett Arena for the first time since leading nearby Collins Hill High School to the last of three consecutive state titles in 2007, and she had strong support from the suburban-Atlanta crowd.
Moore, the WNBA rookie of the year three years ago, was a big reason why the Lynx won their first championship in 2011.
This time, she was named the finals MVP as Minnesota became the second team to go unbeaten in seven playoff games and win a championship, joining the 2010 Seattle Storm.
Lindsay Whalen scored 15 points and Seimone Augustus had 14 for the Lynx, who improved to 14-6 on the road this year.
The Dream cut the lead to three points early in the third quarter, but Moore came off a screen on the next possession and hit a 3-pointer to make it 45-39. She and Brunson combined for 19 of Minnesota’s 26 points in the third.
Atlanta forward Angel McCoughtry, the WNBA’s two-time defending scoring champion, finished with 13 points and shot just 28.5 percent in the finals.
The Dream, forced to move the elimination game to the northern suburbs with their home court booked for an ice skating event, dropped to 6-16 outside Philips Arena.
Minnesota never trailed and made life tough on them from the start. By the 4:52 mark of the first quarter, the Dream had committed five turnovers and was 1-for-5 from the field when coach Fred Williams called a timeout with Atlanta trailing 13-3.
After McCoughtry scored inside to make it 47-42 early in the third, the Dream missed six of their next seven shots and committed four turnovers on their next eight possessions.
Monica Wright followed with a three-point play put the Lynx up by 16.
Atlanta cut the lead to eight early in the fourth, but Devereaux Peters, standing at the top of the key, hit Moore with a backdoor pass for a layup that made it 68-58.