In the ensuing penalty shoot-out, veteran goalkeeper Andy Dibble was the hero, making three successive saves after Stuart Green buried Hull's first attempt.
Darren Ferguson's opener for Wrexham was saved by Matt Duke, but Dean Bennett, Danny Williams and Chris Llewellyn all hit the target.
Wrexham deservedly led 2-1 at half-time but struggled to get out of their own half during the second period and all of extra time as Hull manager Peter Taylor successfully changed tactics to nullify the threat of Hector Sam and Ferguson.
Wrexham exposed flaws in Hull's defence in the early stages and it was no surprise when Sam placed a 14th minute header beyond Duke's reach.
The Red Dragons outpassed Hull at this stage although giving up their lead when Michael Keane diverted an off-target effort from Delroy Facey into goal.
Ferguson curled a free-kick onto the bar and an Armstrong just cleared the woodwork.
Ferguson regained the lead for Wrexham with a powerful shot after 35 minutes and his side went in at half-time apparently in total command.
Hull however took over the momentum as Marc Joseph and Leon Cort started to attack from the back.
Substitutes Andy Dawson and Stuart Elliott made a big impact and it was Dawson's cross that allowed France to level the match after 66 minutes.
Hull kept up the pressure for the rest of normal time and looked the only possible winners in extra time, but Jonathan Walters wasted two gilt-edged chances while Dibble made some great saves.