Giants reliever Dylan Smith then secured his first career major league save by retiring the Cubs in the bottom of the 10th inning, forcing Michael Busch and Bregman to pop out.
"We needed this one, needed to be able to find a way to get it done," said Matt Chapman.
"There was some crazy back and forth, and our bullpen was able to come in and shut it down."
Giants manager Tony Vitello expressed full confidence in sending reliever Keaton Winn to face Pete Crow-Armstrong in the ninth inning, recovering from a blown save the prior day.
Winn successfully forced Crow-Armstrong into a groundout to escape a two-on, two-out jam in the ninth frame.
>>> Jameson Taillon Exits Early as Chicago Cubs Fall to San Francisco Giants
"Yeah, I loved it, I was actually looking forward to it," said Winn.
Giants starting pitcher Trevor McDonald earned a no-decision after giving up one run on four hits over five innings while striking out six.
The defeat drops the Cubs further into a prolonged slump, having gone 0-8-1 in their last nine series to fall 7.5 games behind the Milwaukee Brewers in the National League Central.
Cubs third baseman Alex Bregman finished the night 0-for-5, lowering his season OPS to .
669 and drawing boos from the home crowd after popping out for the final out of the game.
"It's not what any of us expected," said Taillon.
"It's just a matter of what it's going to take to get us back to being that good team.
