The White Sox - who received a letter of support from lifelong fan Pope Leo XIV before their wildcard series at Houston began - fended off a spirited Astros comeback thanks to Meidroth's blast to win 7-3.
"That's everything, man. Coming through for the guys, there's no better feeling in the world," said Meidroth.
Having defied expectations to even reach post-season, after losing 100 games in three dreadful previous years, Chicago progressed from the best-of-three series in just two games, and now faces Cleveland in an American League division series.
The White Sox followed up an impressive 6-3 win Tuesday with an explosive start, racking up four runs in their first inning.
Sam Antonacci hit a double off the game's fourth pitch, and scored off Kyle Teel's ensuing single. Andrew Benintendi's line drive sent Teel over home plate, with Miguel Vargas following him after an overthrow.
Benintendi made it 4-0 from a Tristan Peters single. With bases loaded the Astros yanked ace pitcher Hunter Brown and finally closed the first inning.
Houston did not collapse. Jeremy Pena, Cam Smith and Jose Altuve each walloped solo home runs - in the first, second and fourth innings - to narrow the deficit.
Chicago went without a hit until the sixth.
But, following two walks and three stolen bases, Meidroth emphatically ended that barren run and put the game to rest with a three-run homer over left field.
"I was just looking for something out there. Happened to get a good pitch. You know, he's a really good pitcher. He left one up for me, and I was able to catch it in," said Meidroth.