The League Two side have won their first match in the competition since 2007, winning 4-2 on penalties after the two sides failed to find the net in 90 minutes at the Racecourse Ground.
Charlie Hughes missed first for Wigan, who play in League One, while Wrexham scored all of their kicks.
Thelo Aasgaard blazing his effort over the crossbar sealed Wrexham's triumph, and a buzzing Reynolds was beaming with pride.
"Another year of hearts jumping outta chests like we're in a f****** Aliens movie," the Wrexham co-owner tweeted.
It's a brilliant response from Wrexham, with their return to the EFL not getting off to the best of starts as they suffered a 5-3 defeat at home to the MK Dons last weekend.
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The Red Dragons have sealed their place in the second round of the Carabao Cup which will see Premier League teams not playing in Europe enter the competition, including Chelsea, Tottenham and Everton.
The remainder of the first round ties will take place on Wednesday evening with the second round draw happening after the conclusion of the matches.
The second round ties will take place in the week commencing Monday August 28.