Brendan Rodgers' Bhoys had been knocked off the summit by the Dons' 1-0 win over Dundee United on Saturday, but they responded by easing to their eighth victory in nine games at Fir Park.
Luke McCowan steered in the opener on his first Celtic start, after Lennon Miller had twice struck the woodwork in the early stages.
Stand-in captain Alistair Johnston then ghosted in to score their second at the back post shortly before the hour mark, with Adam Idah adding a third before Motherwall's Liam Gordon was sent off in stoppage time.
That result piled the pressure on Rangers, who could have ended the weekend nine points adrift of the top two, but they claimed a hard-fought 2-1 victory over St Mirren at Ibrox to stay within six.
Mohamed Diomande's powerful close-range finish put the Gers ahead, but Alex Gogic rose to nod in a St Mirren equaliser midway through the first half.
Vaclav Cerny spared their blushes, though, as he scored the winner following good work from Cyriel Dessers in the 69th minute.
Rangers go to second-placed Aberdeen, who are yet to lose this season, for a huge fixture in the title race on Wednesday.
In Sunday's other Premiership game, teenager James Wilson struck three minutes from time to rescue a point for Hearts in a 1-1 Edinburgh Derby draw at Hibernian.
Mykola Kuharevich had put Hibs ahead on the rebound after Marvin Ekpiteta headed against the crossbar just after the hour mark, but Wilson got the final touch to divert James Penrice's long throw home with stoppage time looming.
The result keeps Hibs rooted to the foot of the table with six points, though they are only behind 11th-placed Hearts on goal difference and still have a game in hand to come.
Defender Christopher Jullien tested positive for COVID-19 after the team returned to Scotland ahead of Monday's Premiership clash with Hibernian.
Although all other squad and staff members returned negative tests, Lennon, assistant John Kennedy and 13 first-team players are self-isolating after being deemed close contacts.
The reigning champions remain insistent they followed "rigorous" COVID-secure protocols and that the threat of contracting the virus would have been similarly high had they stayed in Scotland.
In a statement, Celtic said: "Clearly we are hugely disappointed, as we know our supporters will be. The contacts were identified during the period from Wednesday last week, primarily around flight and team coach travel, during which time Celtic applied the same rigorous protocols used for pre-season training camps, UEFA match travel and for all domestic match arrangements in Scotland.
"These protocols have served us well in the past, as the club has not had one positive case in our own 'bubble' until now. As we have already stated, Celtic's decision to travel to Dubai for a training camp was for performance reasons. Whilst we were in Dubai, the announcements made on January 4 significantly changed the COVID landscape.
"The reality is that a case could well have occurred had the team remained in Scotland, as other cases have done in Scottish football and across UK sport in the past week.
"Celtic has done everything it can to ensure we have in place the very best procedures and protocols. From the outset of the pandemic, Celtic has worked closely with the Scottish Government and Scottish football and we will continue to do so."
The hosts took the lead in the 34th minute, with Tom Lawrence whipping a delightful finish into the top-right corner after cutting inside from the left.
Hibs were handed a golden opportunity to equalise on the stroke of half-time, with John Souttar penalised for a handball following a VAR review, but Butland got down to make a brilliant save.
After the break, Cyriel Dessers was denied at point-blank range and Robin Propper twice failed to hit the target in quick succession, but Rangers' lone goal proved to be enough.
While Clement was frustrated by the penalty decision, he was quick to praise his goalkeeper's work as they held on to claim all three points.
"Jack's save was a massive moment," he told RangersTV. "I think Jack was happy as he has been less involved in the last few weeks - it's a good feeling to be important in those moments.
"For me, it wasn't clear why it was a penalty. We discussed these things before with the other managers and the referee department.
"The clean sheet is massive. We brought in Jefte, who is new, Nana Kasanwirjo has come in, and we have a new pairing at centre-backs with John and Robin Propper.
"It is not just defensively; we also have Connor Barron who is new in front of them. They haven't played together too long, but with the organisation, Jack hasn't had to make many saves.
"But when he did, he came up with a massive one and that is a top goalkeeper who keeps his concentration in the big moments."
Rangers came into this game on the back of a Europa League victory away to Malmo on Thursday.
While far from their best, Rangers still managed to claim a fourth consecutive victory in the Scottish Premiership without conceding for the first time since February 2022.
Clement admitted the quick turnaround affected his players' performances.
"The first half we played well, and we deserved the lead, although we could have scored more," he added.
"In the second half, it was more difficult, you could see the freshness going out of the legs of several players.
"We didn't get back to the training centre until 2am on Friday, and then we played at midday, so we had less freshness, so it was important to get the three points and another clean sheet."
Two late goals from Martin Boyle settled a compelling quarter-final that flowed from end to end.
The Paisley side had taken the lead through Scott Tanser, fell behind to quickfire second-half strikes from Elie Youan and Dylan Vente before Keanu Baccus made it 2-2.
Boyle’s double continued Montgomery’s positive start to life at Easter Road and he now has his first working trip to Hampden to look forward to.
He said: “It’s all about getting through to a semi-final at Hampden so I’m really proud of everyone involved.
“I was at Scotland versus England a couple of weeks ago and thought the atmosphere was amazing – apart from the boos!
“But I never went there as a player or coach, so I’m really looking forward to that.
“The boys thoroughly deserved it, so I’m really proud of their effort.
“I thought it was a really good game of football, a good cup tie under the lights.
“I saw great character from the boys and I thought we deserved to win the game.
“We created more than enough chances to win two games. But fair play to St Mirren, they came as the form team in the league.
“They’ve had a fantastic start to the season but we saw that as an opportunity to put an end to that run. I’m really proud of the boys’ effort because it has been a big couple of weeks.
“I thought that performance epitomised everything in terms of team spirit. And in the second half we played some great stuff.”
St Mirren boss Stephen Robinson felt it was a chance missed for his team, who lost for the first time in 11 games.
He said: “It’s an opportunity missed. We were superb in the first half. We had total control of the game and it should have been 2-0.
“But then it was as if we went ‘that was really good, let’s not do that again’ for the first 15 minutes of the second half. Ultimately, we conceded goals that we don’t concede.
“If you lose four goals you’re going to lose football matches. But I can’t be too hard on the players because they showed good fighting spirit and quality.
“Individual errors and bad decision making have cost us. We didn’t deserve to go through but we won’t feel sorry for ourselves. We will reset and come out fighting again on Saturday.”
Hibs dropped back to the bottom of the table after passing up a host of presentable opportunities in Glasgow, with Kasper Schmeichel making several saves in the Celtic goal.
Arne Engels had put Celtic ahead in the sixth minute, with a Joe Newell own goal and a Kyogo Furuhashi strike making the points safe after half-time.
Celtic have not lost a domestic game since a 2-0 defeat to Hearts in early March, and they now hold a nine-point advantage over Aberdeen – who they beat 1-0 on Wednesday – at the top of the table.
Celtic gave up 14 shots worth 2.19 expected goals (xG), while attempting 21 worth 3.89 xG in an open game on Saturday, with Rodgers subsequently acknowledging they were not at their best, particularly in defence.
"It was a great, great win for us on the back of a tough game in midweek," Rodgers said. "You can see where Hibs' confidence has improved and they had chances.
"We were nowhere near the level of our defending, even though we've kept a clean sheet and that's a huge testament to Kasper because we haven't given away much at all this season.
"Today we were just slightly off it and that gave them opportunities to score but, overall, I think to get the win and to keep winning really shows you the mental toughness and the character within the team."
Celtic are back in Champions League action against Dinamo Zagreb on Tuesday, before facing Old Firm rivals Rangers in the Scottish League Cup final next Sunday.
Rodgers' side are into the quarter-finals of the Scottish League Cup after a 3-1 victory at Celtic Park.
Daizen Maeda scored twice in the opening 15 minutes during a one-sided start before Nicolas Kuhn's second-half finish made sure of victory after Mykola Kuharevich's header had the visitors back in the contest.
Celtic will face Falkirk in the next round, where Rodgers will hope his Scottish Premiership champions can replicate Sunday's fine performance.
"I thought we were excellent from the start of the game. We showed our hunger," Rodgers said after the comfortable triumph.
"We said before the game that our playmaker today would be our counter-pressing. It was there and we were winning so many second balls.
"Our only real danger was from throw-ins. They've got guys on either side who can throw it in and add a little bit of pressure and make you fret a little bit.
"We dealt well with that, and their header was a great header. We don't have any complaints. We could maybe have shifted our wall back a bit but we'll learn from that."
This victory came just a week after Celtic had dispatched the same opponents 2-0, their second victory in as many league games to start the season.
The Hoops also hammered Kilmarnock 4-0 in their Scottish Premiership opener, with an away trip to St. Mirren the next test of their title defence.
An Old Firm challenge then awaits as Celtic welcome Rangers on September 1, though Rodgers will be confident of his side's capabilities based on their early showings this term.
