Adam Matthews scored his first goal for Celtic as the Glasgow giants progressed to the fifth round of the Scottish Cup with a narrow 1-0 win over Second Division Arbroath in their replay at Gayfield on Wednesday.
Lassad Nouioui had an early effort ruled out for offside before Welsh defender Matthews finished a lung-bursting run by firing an unstoppable strike past Scott Morrison in the 18th minute.
Steven Doris, who scored the equaliser in the 1-1 draw at Celtic Park 11 days ago, thought he had repeated his heroics when he knocked Colin Hamilton's header over the line from close range in the 63rd minute.
However, in a controversial decision referee Craig Charleston disallowed his effort as he ruled that keeper Fraser Forster had been fouled in the build-up - a ruling that infuriated the team of part-timers.
Substitute Conor Birse was presented with a great chance to send the game into extra time near the end when Doris sent him racing towards goal. However, his effort was turned away for a corner by Forster as the Scottish Premier League leaders progress to meet Raith Rovers.
Celtic manager Neil Lennon said his side's performance had been full of character.
"It was a horrible night and the pitch became a great leveller - it was almost like an ice rink in the second half," said the Northern Irishman.
"It was a real test of character for us tonight but I'm delighted with the attitude of the players. It was uncomfortable at times but that's cup football. We're into the next round and that's another competition to look forward to in the New Year."
Arbroath player manager Paul Sheerin felt there was nothing wrong with his side's disallowed goal.
"My initial reaction was I didn't think it was a foul. I'd have to see it again to really make a decision on it but my first thought was there wasn't much wrong with it," Sheerin said.
"There is a slight frustration but over 180 minutes to match the champions of Scotland the way we've matched them the boys have been excellent and I'm very proud of them tonight."
Celtic had made just three changes from the side that cruised to victory over Kilmarnock at the weekend with Beram Kayal, Georgios Samaras and Mikel Lustig returning to the starting 11.
The Hoops had the ball in the net in the 11th minute when Lassad brilliantly converted a Samaras cross but the Tunisian striker had strayed offside and his strike was ruled out.
Arbroath player manager Paul Sheerin saw his shot from the edge of the box blocked by a Celtic defender before Matthews gave the visitors the lead with a sensational strike in the 18th minute.
The Welsh defender robbed Hamilton of possession on the half-way line and ran into the box before unleashing an unstoppable rising shot into the top right-hand corner of the net from just inside the box.
Celtic should have doubled their lead in the 33rd minute when Samaras set up Lassad for a shot but he fired his weak effort into the turf and Scott Morrison saved it with his knees.
Arbroath felt they should have had an equaliser when Doris knocked home from close range after getting on the end of Hamilton's headed knock-on from a corner but referee Charleston adjudged Alex Keddie had fouled Forster in the build-up.
With just over 10 minutes to go Birse missed a great opportunity to haul Arbroath level when Doris released the substitute with a perfectly weighted pass but his low shot was diverted for a corner by the out-rushing Forster.