Arthur Okonkwo was the Wrexham hero with stops from Igor Jesus and Omari Hutchinson giving them a 4-3 victory on spot-kicks after Josh Windass, Callum Doyle, George Thomason and Jay Rodriguez had converted from the spot.
First-half goals from Liberato Cacace and Oliver Rathbone gave Wrexham early control and had Reynolds trading high fives and hugs from his vantage point in the main stand.
Jesus cut the deficit before Dominic Hyam's diving header restored Wrexham's two-goal advantage with 16 minutes remaining.
Forest looked dead and buried but substitute Callum Hudson-Odoi produced two outstanding finishes, the equaliser coming in the 89th minute.
Wrexham have won three successive promotions to go from non-league to the Championship, but this was the first time since being taken over by Reynolds and fellow actor Rob Mac nearly five years ago they had met Premier League opposition.
Forest made eight changes after beating West Ham on Tuesday, with goalkeeper Matz Sels, Jesus and Hutchinson remaining and Wales international Neco Williams on a stacked bench at his local club.
There was another upset at Deepdale, where Championship high-flyers Preston lost 1-0 to League One Wigan following teenager Harrison Bettoni's strike.
The 18-year-old saw his 75th-minute shot loop into the net via a deflection off substitute Andrew Hughes and, although Callum Wright missed a late penalty, Latics manager Ryan Lowe got one over on his former employers.
Oxford are into the fourth round but only after the Championship strugglers edged out League Two MK Dons on penalties following a 1-1 draw.
Aaron Collins put the Dons ahead before Will Lanskhear equalised for the visitors, and misses from the spot by Jonathan Leko and Connor Lemonheigh-Evans won the tie for Oxford, who are into the fourth round for the first time since 2020.
Jon Brady's first game in charge of Port Vale ended in victory as the Valiants defeated Fleetwood 1-0.
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