Brendan Rodgers still hasn’t lost a Scottish Cup tie and Livingston are yet to savour a first ever win at Celtic Park.

Those statistics, though, tell only part of the story of a compelling cup tie in which the visitors more than played their part before succumbing late on to the third goal of a Daizen Maeda hat-trick that helped steer his team into the semi-finals.

Celtic supporters would have considered this cup clash against the Premiership’s bottom side as welcome respite from their league struggles, but again it was far from straightforward. The home side twice led and were twice pegged back before eventually finding a way to prosper thanks to their persistence and the Livingston defence switching off at inopportune moments.

Daizen Maeda scored a hat-trick as Celtic eventually moved past Livingston in the Scottish Cup.

With Kyogo Furahashi again starting on the bench – he would have his moment late on – Celtic had cause to thank the finishing prowess of another of their Japanese imports. Maeda’s dynamism had been key to 10-man Celtic winning at Livingston earlier in the season and his hat-trick here ultimately proved to be the difference between the sides.

His first goal owed much to the perceptiveness of Nicolas Kuhn who spotted Maeda’s run and then picked him out with a deft left-foot cross that the forward controlled and finished well. Maeda’s second, after Livingston had drawn level for the first time, was a more instinctive effort, the Japanese stooping to steer a header into the net after Matt O’Riley’s shot had been well saved by Michael McGovern. The third – with extra-time just four minutes away – showed his predatory instincts as he poked home Tomoki Iwata’s cross from short range. How Celtic needed that intervention after another patchy performance against a doughty Livingston side who refused to go down quietly.

Celtic were again without Callum McGregor and Cameron Carter-Vickers due to injury, with the latter’s absence particularly felt at Livingston’s first equaliser that halted the home side in their tracks. Cristian Montano’s clipped ball down the line would likely have been mopped up by the ever-alert American international but his understudy, Stephen Welsh, wasn’t as diligent. By the time Welsh had spotted the danger, Daniel MacKay was racing clear before applying an impressive finish beyond Joe Hart.

It is hard to imagine Celtic conceding the goal Livingston plundered for their second equaliser 10 minutes into the second half had McGregor and Carter-Vickers both been on the pitch. O’Riley dallied in possession and was robbed by Jamie Brandon whose quick pass sent Tete Yengi through. O’Riley got back to cover but the striker simply turned away from the midfielder before fizzing a stunning shot into the far corner of Hart’s net.

Celtic Park, for the first but not last time, grumbled in exasperation and it could have been much worse not long after had Michael Nottingham directed his header either side of Hart rather than straight at him.

Livingston threw on another striker in Kurtis Guthrie sensing a chance to win it inside the 90 minutes only to be unpicked again by Maeda who also came within the width of the crossbar of claiming a fourth goal late in the contest. Instead it was his countryman, Furuhashi, who would have the final word, fastening on to James Forrest’s header and finishing well to end any doubts about the outcome.

Celtic Park grumbles return in compelling cup tie as Daizen Maeda's magic moments save the day