Phil Parkinson's men twice took the lead but the Premier League giants took the game to extra-time and with the Reds down to 10-men following George Dobson's dismissal, the visitors went 3-2 up before Wrexham had a goal ruled out following a VAR check.
Chelsea scored a fourth goal late on but they were pushed all the way by Wrexham, and here's five talking points from the game which was watched by Wrexham's co-owners Ryan Reynolds and Rob Mac.
1. VAR spoils a game.
It was the first time that VAR has ever been used at The Racecourse so it was something new for many Wrexham supporters who probably gave it the thumbs down following the defeat.
The Reds took the lead twice against Chelsea and both goals eventually stood following a VAR check, which takes the shine off fans being able to celebrate a goal if you they are unsure whether it will stand or not.
Parkinson felt VAR shouldn't have been used to determine whether Dobson's foul deserved a red card and the extra-time equaliser was ruled out following a check, and although that proved to the correct decision, having the technology in use to aid the on-pitch officials is something Wrexham will have to get used to if they are promoted to the Premier League.
2. Callum Doyle was brilliant once again.
Doyle is having a decent run in the Wrexham side for a reason because he is proving to be one of Parkinson's best signings for the club's return to the Championship.
He is a solid defender who made two vital blocks in his own area to thwart Chelsea but he always proved his worth at the other end of the pitch.
After setting up Sam Smith for the opening goal with a superb pass over the top, Doyle got the final touch to Josh Windass' shot to put Wrexham 2-1 up and he was deservedly named man of the match.
3. What an atmosphere.
The Racecourse was raucous for the fifth round tie against Chelsea as supporters got behind Wrexham from the very start.
They had a lot to cheer about with the Reds giving a fantastic account of themselves on the pitch by more than matching the Premier League giants and fans responded by making it an electric atmosphere.
Unfortunately, Wrexham were unable to take the tie to a penalty shoot-out but supporters stayed to applaud the players off the pitch following the final whistle and it was thoroughly deserved, with fans playing their part in making it a pulsating cup tie.
4. Beaten but not disgraced.
Wrexham didn't get the result they wanted but the players deserve a lot of credit for the way they pushed Chelsea, the current Club World Cup champions, all the way.
They gave everything, were the better team and deserved to at least take the game to a penalty shoot-out following a fantastic performance.
Anyone at The Racecourse of watching on television will agree that the Reds are a very good side and Parkinson is doing a brilliant job.
5. Back to the bread and butter of the league.
Wrexham's cup run is over but there is still so much to play for in the final two months of the season.
Despite not playing a league game on Saturday, the Reds remain in the Championship play-off places and they have 11 matches remaining in the quest to secure a top-six finish.
They are all cup finals now and the run-in continues with a home clash with Hull City - the side immediately above Wrexham - tomorrow night so that is huge for both teams.