In sizzling, 100-F (38c) temperatures at Dodger Stadium, Skubal gave up just two hits, including a solo homer to Sean Murphy in the sixth, and had ample support from the Dodgers offense as Los Angeles seized a 1-0 lead in the best-of-five National League division series.
"I think I was just trying to match the temperature out there," said Skubal, whose fastball hit 101 mph in the second inning. "It was a hot one."
Hernandez gave the Dodgers a 2-0 lead with a 431-foot fourth inning blast off Atlanta relief pitcher Victor Mederos.
Tucker doubled the lead with a two-run shot in the sixth inning with Muncy adding a solo homer two batters later.
Ozzie Albies belted a two-run homer for the Braves to slice the deficit to 5-3 in the ninth inning before the Dodgers closed it out.
The game was one of four second-round series openers on Saturday.
In Florida, Rasmussen held the Yankees without a hit through eight and two-thirds innings.
Yankees catcher Austin Wells broke up his bid for just the fourth no-hitter in MLB post-season history with a double down the first-base line with two out in the eighth.
But Wells's inexplicable attempt to reach third saw him thrown out on a perfect relay from Victor Mesa Jr. to shortstop Taylor Walls to third baseman Junior Caminero.
"Just obviously not a good decision there," Yankees manager Aaron Boone said. "But (the) biggest issue is just Rasmussen shutting us down."
Rasmussen struck out 10 and walked two over eight innings in front of a raucous 40,000-strong crowd at Tropicana Field to make Jonathan Aranda's third-inning home run off Gerrit Cole stand up.
The Chicago White Sox, fueled by a two-run home run from Munetaka Murakami, beat the Guardians 3-0 in Cleveland for a 1-0 lead in their American League division series.