Despite dominating the contest with 70% possession, the Hoops struggled to create clear-cut chances against a lively Kilmarnock outfit, who have come out on top against the Glaswegian giants for the first time since September 2018.
The best two chances of the first half fell to Killie defender Corrie Ndaba, who came close with a second-minute volley before glancing a header narrowly wide in stoppage time.
However, the hosts managed to break the deadlock on the 59-minute mark when Marley Watkins tapped home at the back post after running onto a low-driven cross-cum-shot from Danny Armstrong.
Brad Lyons made a crucial block to prevent Celtic substitute and summer signing Yang Hyun-jun from firing a shot on target nine minutes from time, and a dogged defensive display helped Kilmarnock hold on to their slender victory, sparking jubilant scenes from the Rugby Park faithful.
While Kilmarnock remain on course to reach their second successive League Cup semi-final, 21-time champions Celtic have been eliminated as early as the second round for the just the second time in the last 10 years.
Celtic had never lost a domestic cup match under manager Brendan Rodgers during his first spell in charge, and the Northern Irishman acknowledged after the defeat to Kilmarnock that his Hoops side "weren't good enough".
Speaking to BBC Sportsound, Rodgers said: "We didn't do enough to win the game. We're disappointed with the goal. It comes from a throw-in and, despite being a man down at the time, we should be winning our dual and defending it better.
"We didn't do enough when we had the ball. We didn't show enough personality and didn't show the quality to win the game.
"You have to give all the credit to Kilmarnock. They defended strong, got their blocks in and took one of the few chances they had in the game. I have to analyse ourselves, though, and the quality wasn't good enough in the game."
Kilmarnock boss Derek McInnes has heaped praise on his side's performance, adding: "I'm really pleased with everything I saw today. I thought we limited Celtic to very little. It's testament to the players in front of my goalkeeper, who, across three games, has faced two or three shots.
"To beat such a top opponent at this stage of the season, you have to do a lot right. We dealt with the changes Celtic have the luxury of making really well and I'm delighted we're through to the next round of the cup tie.
"There's no point in beating Celtic if we don't go on to win it now. I've just said that to the boys. No one will remember this result if we don't go on and win it."
Kilmarnock have been rewarded with a quarter-final home draw against Hearts, who thrashed Championship side Partick Thistle 4-0 at Tynecastle on Sunday.
Rangers, who came from behind to beat Greenock Morton 2-1 on Saturday, have been drawn at home against Livingston, who secured a routine 2-0 second-round win over Ayr United.
After beating Airdrieonians 4-3 in extra time on Saturday, Ross County have been drawn at home against Aberdeen, who edged past Stirling Albion by a 2-1 scoreline, while Hibernian and St Mirren will lock horns at Easter Road after the two teams progressed from the second round with wins over Raith Rovers and Motherwell respectively.
All League Cup quarter-final ties are scheduled to be played during the midweek of September 26-28.
Scottish League Cup quarter-final draw in full:
Kilmarnock vs. Hearts
Rangers vs. Livingston
Ross County vs. Aberdeen
Hibernian vs. St Mirren