Kershaw elevated a 2-2 fastball to lead off the third inning and Woodruff didn't miss it, belting a towering shot that went over the fence in right-center field at Miller Park to tie the Los Angeles Dodgers 1-1.
It marked only the fourth time — in more than 2,228 career innings — that Kershaw had given up a home run to the opposing pitcher, and the first time in the playoffs.
"It's something, you know obviously coming into the day you don't know in your wildest dreams that that's going to happen," Woodruff said after the Brewers held on to win 6-5. "You know, to be able to get an at bat off Kershaw and hit a home run. But, I rounded first, and once I knew it was gone, it was just one of those, kind of moments you just, you're not really thinking, and I was just letting some emotion out. It was a cool moment, and I was happy I could just go out there and do it for the team."
Woodruff, who relieved Brewers starter Gio Gonzalez after two innings, became the third reliever to hit a home run in postseason history, according to ESPN Stats & Information research. The 25-year-old's home run was his second in 19 career at-bats.
"I knew he could swing the bat a little bit for sure," Kershaw said. "I don't know he could do that. But I knew he could hit a little bit."
Woodruff, who also earned the win after pitching two perfect innings, became the first lefty-hitting pitcher to go deep against a fellow left-hander in the playoffs, and the first pitcher to homer against a former Cy Young Award winner since the 1970 American League Championship Series.
"This guy's a good athlete, don't sleep on him," Ryan Braun told Fox Sports. "But that's not something you ever expect [from] any pitcher, especially against a guy as good as Kershaw."
The Dodgers took the initial lead on a home run by shortstop Manny Machado that had an exit velocity of 115.6 mph. But Kershaw struggled through a third inning that ultimately saw him throw 29 pitches, later yielding a sacrifice fly to Hernan Perez.
Woodruff's spot came up again in the bottom of the fourth, but with runners on second and third, Brewers manager Craig Counsell opted to pinch hit for him.
Domingo Santana drove in both runners with a single to left, putting the Brewers up 4-1 and chasing Kershaw from the game, making it the shortest postseason start of his career.