Michael Chopra, with his first goal in 14 matches, looked to have booked Cardiff a home tie with Leicester City, but Bristol substitute Gavin Williams grabbed a stoppage-time equaliser.
A poor goalkick by Dean Gerken opened the way for Chopra's tight angled shot in the 76th minute, but a desperate run by centre-back Liam Fontaine to the byline set up Williams in the 92nd minute.
The crowd of 7,289 - half the average league gate at Ashton Gate - peered through almost non-stop snow.
Three hundred tonnes had been cleared off the pitch and a heated dome had done enough for it to pass the inspection of referee Darren Deadman.
Snow came back three hours before the kick-off and the slippery, slushy surface soon made possession a lottery.
Five minutes in, home captain Louis Carey powered a header from a corner against the woodwork only to earn a rebuke from Deadman.
The home side's full-backs, Bradley Orr and Jamie McAllister, were not so fortunate with both being booked in the opening quarter.
The first free-kick from 50 yards drew an optimistic shot from Ross McCormack while the second taken by Anthony Gerrard from 40 yards almost put Cardiff ahead.
Gerken pulled down Gerrard's shot but then had to make a sharp follow up save from Chopra.
Conditions soon made it a scrappy tie with centre-back Carey looking the most likely player on either side to make a breakthrough.
Cardiff manager Dave Jones was so short of players he couldn't fill all the seats on the bench and his problems mounted just before half-time when Gavin Rae limped off with a strained hamstring.
With Bristol offering little in attack, manager Gary Johnson made a voluntary change for the second half, sending on Alvaro Saborio for the ineffectual David Clarkson.
Saborio flashed one shot wide but Bristol then had to repel a flurry of corners from Peter Whittingham before heading for next Tuesday's replay.