The Premier League giants twice trailed at a raucous Racecourse but battled back to reach the quarter-finals with a breathless 4-2 success after extra time.
Sam Smith and Callum Doyle scored for Wrexham but the visitors took control after George Dobson was sent off for a high challenge on Alejandro Garnacho, who then scored along with Joao Pedro to settle the tie.
Reds' boss Phil Parkinson disagreed with the decision to use VAR to check Dobson's foul, saying referee Peter Bankes shouldn't have gone to the off-field monitor for a review.
Wrexham thought they might force penalties when Lewis Brunt found the net at 3-2, but the effort was ruled out for offside following a VAR check, and Chelsea progressed to the quarter-finals.
Here's what supporters thought about the use of VAR at the game.
PHILIP IAN HARVEY: "Offside due to the width of a toenail. VAR and the offside rules really need sorting out, not just because of last night, but in general. We need to get some common sense back into the game."
MIKE GREEN: "Clear and obvious mistakes is when VAR is supposed to come in. If it takes 2/3 minutes to come to a decision then it's not clear and obvious. Needs binning as it's sucking the life out of football."
JON ESPOSITO: "I agree that “something” needs to be done. My question for everyone is, what is the right solution? I tend to think that maybe only the on-field ref should be able to request VAR’s help (not VAR telling them something). But I honestly don’t know. Or, is the problem not with VAR, but with the offside rule? Should it be adjusted?"
JOAKIM DREYER JOHNSEN: "The problem with this is that the referees makes too many wrong decisions. They need VAR to prevent this. The game is moving fast and he can't see everything. It also give all the cheaters a harder time to trick the referee. The only thing I think should be changed regarding VAR is the amount of time they use to look at situations. Lets say they get one minute max and if the original decision of the referee can't be disproved, his decision stands. I want a fair game without wrong goals and cheaters."
PATRICK MARSHALL: "Shearer said that Dobson's tackle was high. He forgets he kicked someone in the head while they were on the ground."
JOHN PARRY: "Some clubs asked to use the technology that's available to get arguably the right decision, when it goes in your favour its great but against its bad, take a look at the Lewis Brunt's face when the goal was cancelled."
BRENT ROBERTS: "Dobson got sent off because of the over reaction of the Chelsea player rolling around as if he had a career changing injury. It should have been a yellow for Dobson and a yellow for the Chelsea player."