The Spurs were always in the ascendancy at AT&T Center after racing to a 31-14 first-quarter advantage, with Kawhi Leonard and Danny Green also making notable contributions of 18 and 16 points respectively.
Back-to-back three-pointers from Quincy Pondexter helped the Grizzlies cut the deficit to six points in the third quarter, but San Antonio swiftly pulled away again with an 11-1 rally in the fourth.
Pondexter hit five 3-pointers and ended the game with 17 points.
Memphis' Zach Randolph had a career-low playoff performance by going 1-for-8 for two points in 28 minutes. He was averaging 19.7 points, 51-percent shooting and 14.7 shots per game in the playoffs.
"It was just one of them nights," Randolph said.
"I've been double-teamed the whole season. A lot of my shots were off. I didn't get as good a look as I wanted to.
"It was just the rhythm of the game. I've got to be better, as I told my teammates. I've got to be better for them."
The Grizzlies also dropped Game 1 in the first round against the Clippers and the conference semis against Oklahoma City.
The best-of-seven series resumes with game two in San Antonio on Tuesday night.