On-form striker Gary Hooper opened the scoring in the 31st minute with a low drive before Well defender Adam Cummins put Scott Brown's cross past Darren Randolph and into his own net four minutes later.
Cummins was sent off soon after the break by referee Iain Brines for a 'last man' challenge on Kris Commons but Brown saw his second penalty in a week saved before a sterling display by Randolph - along with some poor finishing - prevented the scoreline reflecting the Hoops' dominance.
Neil Lennon's men will travel to Russia on Sunday for their Champions League Group G game against Spartak Moscow on Tuesday buoyed by their performance in contrast to the beleaguered Lanarkshire side, who barely troubled Hoops keeper Fraser Forster, who had one of his easiest afternoons.
Stuart McCall's men, knocked out of the Scottish League Cup third-round by Rangers at Ibrox in midweek, showed three changes with Jamie Murphy, Simon Ramsden and Zaine Francis-Angol in for Henrik Ojamaa, Fraser Kerr and Steven Hammell, the latter out of the squad altogether with an injury.
Brown, Forster, Mikael Lustig, Commons and Victor Wanyama all came back after missing the 4-1 Scottish Communities League Cup third-round win over Irn-Bru First Division side Raith Rovers on Tuesday.
Efe Ambrose, who performed so impressively in the middle of the park against Raith in his first start for Celtic, partnered Kelvin Wilson in the heart of the Hoops defence, with Charlie Mulgrew moving wide left in midfield.
Celtic were pushed back in the early stages but it was Randolph who had to make the first real save of the game in the ninth minute from Hooper's close-range shot, after the Parkhead striker had raced on to a well-timed pass from James Forrest.
The visitors, playing in a three o'clock kick-off away from home on a Saturday for the first time in seven years, appeared to turn on a switch and moments later Randolph did well to parry Brown's low drive before Wanyama failed to get the rebound on target.
Commons looked in the mood and he had three efforts on goal, all of which narrowly missed the target, but the third of which was a clever lob in the 22nd minute which almost caught out Randolph.
In the 27th minute, though, it was casual play by the former Nottingham Forest and Derby player which gave possession away to Tom Hateley but the Fir Park defender blazed his shot from distance high over the bar.
Moments later, Hooper looked favourite to beat Randolph to Wanyama's through ball which beat the Well defence, again caught square, but the Well keeper race from his line to clear.
As Celtic kept up the pressure it took a good blocking tackle by Francis-Angol to prevent the Englishman's close-range shot again testing the Steelmen's number one.
However, Hooper was not to be denied and just after the half-hour mark, Commons' perfectly-flighted pass evaded Hutchinson and the former Scunthorpe striker took the ball in his stride before confidently steering it past Randolph for his sixth goal in three games.
Four minutes later Cummins, in his bid to cut out Brown's cross from the left, after the former Hibs player had been picked out by Emilio Izaguirre, knocked the ball into his own net from six yards.
Celtic started the second-half with unsurprising confidence, Forrest playing in Hooper whose driven cross from wide on the right was begging for a team mate to tap into the net.
In the 54th minute the home side's chances of an unlikely comeback disappeared when Cummins was shown a straight red for bundling Commons to the ground inside the box as the Celtic player shaped to go through on goal after racing on to a Wanyama pass.
Brown, who had a penalty saved by Robert Douglas in last week's win over Dundee, stepped up again and Randolph guessed right to parry the shot with Wanyama putting the rebound over the bar.
It was all rather irrelevant though.
Motherwell's 10 men were penned inside their own half, struggling to survive a Celtic siege and in the 65th minute Randolph had to make a great save from Brown, who had latched on to an Izaguirre cross.
Two minutes later, the Scotland midfielder was replaced by Georgios Samaras, who returned for the first time since injuring his elbow on international duty for Greece earlier in the month, before Wanyama headed over another good Izaguirre cross from the left.
Celtic were relentless. Randolph saved from Izaguirre, who decided to have a go himself, before making another block from Forrest, who had outstripped the bedraggled Well defence.
Randolph prevented further damage being inflicted by a rampant Parkhead side who will travel to Russia with confidence.