Murray, traded by the Spurs in the off-season, scored 25 with nine assists and eight rebounds, followed by team-mate Trae Young who had 22 points.
Giannis Antetokounmpo led the Bucks with 19 points but the Greek star, who drew screams from the crowd whenever he touched the ball, could not inspire a turnaround as Milwaukee trailed from the second quarter.
The Gulf's first meeting between NBA teams was held at a packed Etihad Arena in the United Arab Emirates capital, which will also host game two on Saturday.
"We've got to keep that hunger mentality to work... to continue to learn and get better and grow with each other and compete at a high level," said Murray.
"And I believe we can do that... I'm already on to Saturday," he added. "We got the win, but like I said, there's always room to improve."
The Abu Dhabi games are part of a resumption of NBA overseas trips after a shutdown for the pandemic. The Golden State Warriors won two pre-season games against the Washington Wizards in Japan, while Paris and Mexico City will host regular-season fixtures.
The wealthy Gulf is attracting an increasing number of major sports events including this year's World Cup in Qatar and four F1 races next season, as well as golf, tennis and heavyweight boxing.
"It's been a long time coming," NBA Commissioner Adam Silver told reporters before the game, referring to a visit to the region.
"We believe we've had, in the history of the NBA, roughly 25 players from the Middle East, North Africa region.
"It's our hope that as we continue to play games here and televise games... that it will encourage even more young people, frankly boys and girls, whether aspiring to play in the NBA or the WNBA, to continue playing the game."
This week's games are part of a multi-year partnership between the NBA and Abu Dhabi.
The Bucks were named as favourites for this season's title in the annual survey of the league's 30 general managers.