Neil Lennon's side have now won eight games in succession since losing to Livi in West Lothian last month, their only domestic defeat this season.
Rangers have the chance to close the gap at Hamilton on Sunday, while Lennon's men will prepare to take on Rennes in the Europa League on Thursday, safe in the knowledge that his side are already through to the last-32 of the competition and in fine form.
Celtic had gone three games without a win over Livi and without scoring.
Home fans were happy to see fit-again striker Leigh Griffiths, whose last appearance was at the end of August due to personal problems and injury, back on the bench.
Celtic were finding it difficult to get behind the West Lothian side's defence but when Edouard did he scored with his trademark composure.
Gary Holt's side were looking for an offside flag when Rogic sent the Frenchman racing clear but that never came and despite the attention of chasing Livi defender Ricki Lamie, the 21-year-old dinked the ball from 14 yards over Sarkic.
Celtic relaxed and grew in confidence.
Edouard's drive from the edge of the box in the 34th minute, after Forrest had cushioned a searching pass from Rogic straight to his feet, was saved by Sarkic, who made a much better block from the Parkhead forward's powerful drive four minutes later after he had been again set up by the Scotland winger.
The home side finished off the half with Rogic blazing a cut-back from the tireless Frimpong over the bar.
Seconds later, with the Hoops fans encouraging Brown to shoot from 20 yards when he took control of a Christie pass, the veteran midfielder duly obliged and the ball took a deflection before leaving Sarkic stranded.
Forrest's first goal came after Frimpong's angled-drive was saved by Sarkic, the ball spinning high into the air and the Hoops winger despatching it from 10 yards when it descended.
That further cushion allowed Lennon to replace Edouard with Griffiths who got a huge cheer from the home support who subsequently gasped when he could not connect properly with a Frimpong cross in the 78th minute.
Sarkic then made a great top-corner save from Christie before Forrest raced through on to a pass from the former Inverness midfielder in added time to slot in his second and put the pressure on their Old Firm rivals for Sunday.