The Wales international is in the last few months of his contract with his future subject to much speculation but in the 39th-minute of a game played in relentless rain at Parkhead, he showed why Hoops manager Neil Lennon is so keen to keep him by driving in brilliantly from 25 yards.
The victory keeps the Hoops 11 points clear at the top of the Scottish Premiership.
The two teams immediately behind Celtic also picked up victories on New Year's Day with second-placed Motherwell downing St Johnstone 4-0 and Aberdeen earning a 2-1 win at Dundee United.
John Sutton opened the scoring for Motherwell with his 10th of the season following an incisive cross from Iain Vigurs, who then made it 2-0 just after half-time with his first for the club.
Lionel Ainsworth fired home his fourth goal in as many games from 20 yards to increase the lead and the home fans were treated to a special strike from substitute James McFadden, who curled a right-footed shot into the corner of the net from almost 30 yards to make it 4-0 as Stuart McCall's men made it five league wins on the trot.
At Tannadice, Peter Pawlett's last-gasp winner gave Aberdeen the points.
The Pittodrie side were marginally better in a disappointing first half but allowed United to grab the initiative when John Souttar benefited from Jamie Langfield's mistake to head the Terrors ahead after 50 minutes.
Barry Robson levelled the scores with a spot-kick 15 minutes later before Pawlett teed up a dramatic finish with his header beyond the excellent Radoslaw Cierzniak in the 90th minute to ensure his team stay just two points behind the Steelmen in the standings.
Inverness were unable to make it a clean sweep of victories for the top four as they crashed to a surprise home defeat at the hands of Highland rivals Ross County.
Ross had picked up just one point from their previous nine Scottish Premiership games heading into the derby but they were finally able to celebrate a victory thanks to first-half goals from Scott Boyd and Alex Cooper.
Aaron Doran pulled a late goal back for 10-man Inverness - who lost Ross Draper to a red card with 15 minutes to go - but they could not find an equaliser as they fell further adrift of the top three.