Duncan and Parker combined to score seven points in the final 4 minutes to preserve the victory.
San Antonio also forced six turnovers in the fourth quarter and held Utah to 11 field goal attempts after allowing the Jazz to make 12 of 24 shots in the third.
"Our guys were really embarrassed by that," Spurs coach Gregg Popovich said. "Together playing good individual and team defense; holding them to 21 (points) was the key and they knew it."
It was the same defensive intensity San Antonio displayed in building the 19-point lead.
After Gordon Heyward made a 22-foot jumper on the team's first attempt, the Jazz missed their next nine shots as the Spurs raced to a 15-5 lead. Derrick Favors' dunk snapped Utah's dry spell with 5 minutes left in the first quarter.
"We were rolling," Green said. "We were playing well, and we were moving the ball and things were going well. They were playing pretty good defense, and that made us play better defense, which generated our offense and gave us a little boost and a little energy."
The Spurs had seven turnovers in the first quarter, but offset that by holding Utah to 32 percent shooting.
After having only one turnover in his first two games, Parker had three in the first quarter alone.
Ginobili made his season debut with 3:06 left in the first quarter, drawing thunderous applause from the sold out crowd. Ginobili scored his first points in typical style, driving from behind the three-point line for a scooping left-handed layup. His entry in the second half drew an equally loud response from the crowd.
NOTES: San Antonio holds a 63-20 record at home against the Jazz and lead the overall series 91-73. ... The Spurs eclipsed their previous high for points in the first half (43) set in their season opener against New Orleans. The Spurs' record for field goal percentage in a half is 84 percent, which they set in the first half against Cleveland on Dec. 11, 2004.