Billingsley, set to be the Dodgers' No. 2 starter, gave up four hits in six shutout innings to help Los Angeles beat the Chicago Cubs 2-1 in 10 innings.
He ran into trouble in his final inning, giving up a hit and two walks before a sliding catch in center field by Xavier Paul bailed him out.
Billingsley threw 94 pitches.
"And I still felt strong," he said. "I had good fastball command, and my changeup was really good. ... I want to build off today and take it into the season."
Dempster, the Cubs' Opening Day starter, continued his excellent spring by giving up one run and six hits in six innings. He struck out six and walked one, lowering his ERA to 1.88.
"My body feels good. My pitches feel good," Dempster said. "I was able to locate well today.
"I'm looking forward to getting out there again five days from now and tuning up for Opening Day."
The Dodgers won it in the bottom of the 10th when Orlando Mercado lofted a bloop single to left off Jeff Samardzija with the bases loaded and two outs. The hit, which barely eluded a diving Fernando Perez, scored Kyle Russell, who had doubled.
Jeff Baker hit a solo homer for the Cubs in the ninth to tie it at 1.
Eugenio Velez, a non-roster invite to Dodgers camp, sprained his right ankle when he landed awkwardly on first base while running out an infield hit.
The injury probably ends the chance for Velez, who can play the infield and outfield, to make the club coming out of Arizona.
"He was in the mix," manager Don Mattingly said. "He switch-hits, has that speed and is able to play the infield and outfield. He's an interesting guy."
Hector Gimenez, making a long-shot bid for a roster spot with the Dodgers, made his first appearance of the spring in the outfield. He threw out Tyler Colvin trying to score from second on a single by Marlon Byrd. Gimenez, a first baseman, has appeared in the outfield only three times in his minor league career. At the plate, though, he was 0-for-4 with three strikeouts. That lowered his batting average to .294 with three homers and seven RBIs. ... Neither team made an error. Coming into the game, the Cubs had committed 34 errors this spring, second-most in the majors behind San Diego (36). ... The Dodgers reassigned RHP Oscar Villareal to minor league camp. ... Byrd went 2 for 5, which actually lowered his average to .478.