The Premier League champions have now suffered four defeats in their last six games in all competitions and could be without Costa for their crunch clash against Liverpool this weekend after he was reportedly taken to hospital with a chest injury.
Without the Spaniard, the Blues pushed hard for a winner in extra-time at the Britannia but could not find a way past Butland, who made a sharp late stop to keep the game level.
Both sides named strong starting line-ups for the clash, with Hazard, Costa and Oscar in the XI for Chelsea and Ryan Shawcross making his first appearance of the season for Stoke.
The Blues started brightly and Butland was called into action early to keep out a near-post effort from Costa and a shot from Oscar.
Ramires should have given the visitors the lead when he rounded Butland after half an hour but failed to hit the target with the goal at his mercy.
Costa was then forced off holding his side with an injury that looked to be sustained during a challenge with Shawcross and Charlie Adam.
Stoke’s best chance of the half came at the end as Walters turned in the box but saw his shot blocked by the outrushing Asmir Begovic.
The Potters forward would not be denied for long as, shortly after the interval, he was teed up on the edge of the box and sent a fantastic strike crashing in off the underside of bar.
Kurt Zouma and substitutes Kenedy and Bertrand Traore were unable to convert equaliser attempts, and Walters also missed a couple of chances to add to his effort.
Stoppage time then saw things dramatically turn in Chelsea’s favour, with Remy rifling in Zouma’s knock-down from a Willian corner and Bardsley being shown a second yellow card after a tackle on Kenedy.
The first half of the additional period saw Chelsea threaten through Hazard and a Traore header, while Marko Arnautovic fired a shot just wide of Begovic’s post.
Willian went close in the second 15 minutes and Butland saved from Traore with his feet before touching the ball away as it skidded goalwards off Kenedy.
The England goalkeeper then proved the hero of the night as he made the decisive intervention in the shootout to send Stoke through to the quarter-finals.
Source Skysports