Van Slyke and Butera homered in the 12th inning, and Los Angeles held on to beat the Minnesota Twins 4-3 to earn a sweep of the day-night doubleheader Thursday.
Adrian Gonzalez also homered for the Dodgers, who stranded 16 runners in the 5-hour, 11-minute game.
"At this time of the night, when you've played all night and you've put a couple wins under your belt, you're willing to take them, move forward and know that you got to play better to get where you want to go," Dodgers manager Don Mattingly said.
Puig tied a career high with four hits in Los Angeles' 9-4 Game 1 win and added two more in the nightcap. Before grounding out in the fifth, Puig had reached base in nine consecutive plate appearances.
Joe Mauer had three hits for the Twins, who walked 12 in the second game and 18 total.
Van Slyke led off the 12th and lined a 2-1 pitch from Brian Duensing (0-1) into the bullpen in left-center field. Butera hit his first homer since July 12, 2012, when he played for the Twins.
"It feels nice to hit a home run any time," said Butera, who caught all 12 innings and still had enough in the tank to hit the winning homer.
"It's mentally and physically taxing," he continued. "You don't want to make any mistakes. You want to be in the game physically as much as you are mentally."
Jamey Wright (2-1) didn't allow a run in three innings of relief.
Kenley Jansen loaded the bases in the 12th and allowed Mauer's sacrifice fly but got Chris Colabello to line out to first for his 11th save in 13 chances.
"I didn't know if it was going to take off," Colabello said. "It was pretty true. I didn't really side-spin it at all, maybe hoping that it was going to get over his head or get a deflection, or something like that."
Juan Uribe had four hits and two RBIs in the first game and another RBI in the second.
When Puig wasn't getting hits, he still grabbed the spotlight.
The Cuban outfielder was tagged out when he flinched toward second after beating out an infield single in his first at-bat in Game 2. In the bottom of the first, Puig airmailed the cutoff man on a sacrifice fly, allowing two Twins runners to advance.
Plate umpire Tim Timmons missed a catcher's interference call when Puig fouled off the first pitch during a key at-bat in the sixth.
But by the time Van Slyke homered, Puig's play from earlier in the day seemed like ancient history.
"Scotty's a different cat. He's hard to read from the outside," Mattingly said. "You see a guy that's kind of laid back and kind of easy moving. Everything looks like it's in slow-mo. But he's a guy that sees pitching and knows what he's doing at the plate."
Los Angeles left the bases loaded in the sixth and seventh, with the threat in the seventh ending after second baseman Brian Dozier dove to snare a line drive from Dee Gordon.
Neither team's inexperienced starter, both called up to start under a provision that allows rosters to expand to 26 for a doubleheader, finished the fifth inning.
Minnesota's Kris Johnson lasted 4 1/3 innings in his Twins debut.
Wearing his powder-blue socks high and a pink band around his head to keep his glasses on, Red Patterson was removed after allowing one run and walking three over 4 2/3 innings.
Minnesota took a 2-1 lead in the sixth when Trevor Plouffe reached second on Uribe's throwing error and came home on another throwing error as reliever Brandon League rushed to try to retire Colabello on a dribbler to the mound.
The Dodgers tied it when Van Slyke led off the sixth with a triple and scored on Uribe's single.
Minnesota center fielder Aaron Hicks banged his head on the wall leaping to try to catch Van Slyke's triple and left the game with concussion-like symptoms.
The pitching wasn't great in Game 1, but it was much better than in the never-ending night game.
Dan Haren (4-0) struck out seven in 6 2/3 innings to win the first game.
Mike Pelfrey (0-3), who hasn't won at Target Field since April 16, 2013, gave up five runs and seven hits and walked three in four innings for the Twins.
If Pelfrey doesn't pitch better soon, he could wind up in the bullpen.
"We've got to do something about it, we have to fix it," Twins manager Ron Gardenhire said. "We're trying to get him on the right path."
Both teams combined to use 20 pitchers.
Dodgers ace Clayton Kershaw, on the disabled list since after his first start of the season with strained shoulder, threw 86 pitches and allowed two runs in five innings in a start for Double-A Chattanooga on Wednesday night and appears in line to rejoin the rotation early next week. ... Ricky Nolasco (2-2, 6.67 ERA) starts for the Twins on Friday against Baltimore and Ubaldo Jimenez (0-4, 6.59) in the first game of a three-game series. ... The Dodgers continue their nine-game trip Friday in Miami as Josh Beckett (0-0, 2.45) faces Tom Koehler (2-2, 2.97).