Bumgarner became the first pitcher in major league history to hit two homers on Opening Day, but Arizona washed that away by scoring two runs with two outs in the ninth inning for a 6-5 victory over the Giants at Chase Field in Lovullo's first game as manager.
"It's something I'll remember for the rest of my life," an emotional Lovullo said. "It couldn't have been scripted any better. The walk that we've been walking, the concepts that we've laid on these guys, they went out there and fought and earned every part of that victory."
Arizona shortstop Chris Owings singled home the winning run off new Giants closer Mark Melancon, who was signed to a four-year, $62 million contract this winter to shore up a bullpen that struggled in 2016.
Jeff Mathis doubled with two outs in the ninth inning, his third hit of the game, and pinch hitter Daniel Descalso tied the game at 5 with a single to center. A.J. Pollock followed with a single to center to put runners on first and third and bring up Owings.
"I faced him a couple of times in the past, and that really helped out," Owings said. "I was just trying to get something up in the zone. Just staying with it, keeping confidence up. It's tough not to go up there and compete when all your teammates are doing it, too.
"Just trying to fit in."
Bumgarner hit bases-empty homers in the fifth and seventh innings, the second giving the Giants a 4-3 lead. He struck out 11, gave up six hits and three runs and did not walk a batter in seven innings.
The D-backs tied the game in the eighth on consecutive singles by Pollock, Owings and Paul Goldschmidt off Derek Law, the first of two blown saves by the Giants. The Giants led the majors with 30 last season. They had two more in the postseason.
"We never expect anything negative to happen with Mark out there," Giants manager Bruce Bochy said. "It happened. You are not going to be perfect. Sure you hate to lose the lead there, but 'Bum,' what a day he had."
Bumgarner retired the first 16 he faced before Mathis tripled, pinch hitter Nick Ahmed singled and Pollock hit a two-run homer to tie the game at 3 with one out in the sixth inning.
Bumgarner has 16 career homers, three in 2016 and five the year before. He joined Barry Bonds (2002), Matt Williams (1994) and Willie Mays (1964) as the only Giants to homer twice on Opening Day in the past 53 years.
"That is obviously pretty special to do that and for that to happen, but my job is out there on the mound. That is where my concern is at," Bumgarner said.
"We work on our hitting a lot here and sometimes the pitcher's at-bats can decide the game. It had a chance to today but it didn't. That is something you look back when you are done playing and say you did it."
Pollock homered, and he and Mathis had three hits apiece.
Eduardo Nunez had three hits, stole two bases and drove in a run for the Giants. The series will resume Tuesday after an off day while the NCAA basketball championship game is played in suburban Glendale, Ariz.
Nunez opened the second inning with a chopping infield single to third base, stole second, took third on a groundout to the right side and scored on Joe Panik's sacrifice fly.
After Bumgarner homered in the fifth, the Giants took a 3-0 lead when Brandon Crawford doubled and scored on Nunez's single in the sixth inning.
Arizona starter Zack Greinke gave up two runs and four hits in five innings.
NOTES: Giants LF Jarrett Parker made his first Opening Day start after making the roster out of spring training for the first time. Giants LHP Ty Blach, OF Gorkys Hernandez, RHP Derek Law and OF Chris Marrero also made their first Opening Day rosters. ... D-backs manager Torey Lovullo had approximately 75 family and friends at the game. He managed the final 48 games of the 2015 season in Boston when Red Sox manager John Farrell was on medical leave. ... San Francisco manager Bruce Bochy received a nice hand during pregame introductions. The Giants spend spring training up the road in Scottsdale, Ariz. ... Former NBA player Grant Hill threw out the ceremonial first pitch.