The Pilgrims got off to the worst possible start when Conor Hourihane picked up two yellow cards inside the opening 10 minutes.
Exeter loanee Gosling, released by Plymouth aged 16, converted from close range, after a left-wing cross had been flicked on by long-serving defender Mark Jermyn, to book the Blue Square Bet South side a second round clash at fellow non-league outfits Luton or Nuneaton.
Only a late strike from Calvin Zola prevented another surprise result as Burton snatched a 3-3 draw at home to Blue Square Bet North side Altrincham.
Albion are play-off contenders in League Two and although Altrincham striker Damian Reeves saw his fourth-minute spot-kick saved, they still took a 2-0 half-time lead thanks to Carl Rodgers' strike and Nathan Stanton's own goal.
Zander Diamond halved the deficit and Robbie Weir missed a penalty before Jacques Maghoma levelled proceedings.
Burton still looked to be heading out when defender Damien McCrory put through his own net in the 86th minute but Zola spared their blushes to ensure a replay, with Crewe awaiting the winners.
Gloucester City and Leyton Orient's clash as well as Braintree's encounter with League One leaders Tranmere both fell foul of the weather, with Alfreton and Chesterfield awaiting the respective winners.
New dates for the rearranged games are yet to be announced.
Source: PA
Source: PA