Hart, who shouldered the blame for Feyenoord's opener in a midweek loss that saw Celtic finish with nine men, this time saw red for a professional foul on Mo Sangare before the break.

But Livingston could not make their numerical advantage count and Matt O'Riley tapped in Celtic's second before a late stunner from Daizen Maeda.

The defending champions move to 16 points from a possible 18 and will be top at the end of the sixth round of fixtures. Livingston's loss was their second in the league this season and they remain on six points.

Four days on from their Champions League defeat in Rotterdam, Celtic were out to reassert domestic dominance but almost fell behind early on. Bruce Anderson forced a save from Hart, with Scott Pittman's subsequent attempt cleared off the line by Callum McGregor.

Brendan Rodgers' side upped their intent and Hatate raced through to meet Greg Taylor's pass, only to be fouled by De Lucas. The Japanese midfielder's spot-kick squirmed under the body of Shamal George for a first goal of the season.

It was crucial the hosts kept themselves in touch and George made a fine save to deny Maeda.

Hart's game came to an end after Sangare chased a long ball and was clattered by the goalkeeper outside the area. James Forrest made way for reserve keeper Scott Bain.

The 10 players saw out a frantic end to the first half and doubled their lead early in the second. Kyogo Furuhashi's cross should have been converted by Maeda, with the ball coming back off George for O'Riley to finish.

Bain almost gave Livi a route back into the game when he was robbed by Joel Nouble right in front of goal, but the towering forward contrived to chip over the bar.

George again saved from Maeda, and Celtic substitute Oh Hyeon-gyu fired narrowly wide, but a third goal arrived in added time when Maeda curled into the top left corner from the edge of the box - his first goal of the season.

Player of the match - Matt O'Riley

Another solid - and scoring - appearance for the ever more influential Celtic midfielder

Depleted Celtic back to winning ways - analysis

Defensively, Livingston's main task was staying in the game. Going forward, decision-making was always going to be key. On both counts, they fell short.

Without captain Mikey Devlin at the back, the home side were pulled left and right by Kyogo and Maeda and the Livi forwards only fleetingly troubled either of the keepers they faced.

While Tuesday was a reminder of the levels Celtic must try to meet to compete in the Champions League, the forcefulness and resilience they showed in West Lothian demonstrated just how hard domestic opponents must work to trouble the league leaders.

The disruption caused by the Hart incident - Celtic's third sending-off in two games - turned the contest into a much scrappier one for Rodgers' side. Regardless, they passed the test.

What's next?

Livingston visit Rangers in Wednesday's Viaplay Cup quarter-final (19:45 BST). Celtic are next in action away to Premiership opponents Motherwell next Saturday lunchtime (12:30).