The home side came close through efforts from Nick Ross and Josh Meekings before Georgios Samaras took advantage of a Ross Tokely mistake to send a scorching shot past stand-in keeper Jonny Tuffey in the 33rd minute for his first away goal since January 2011.
The Greek striker missed a handful of chances to double the lead before Gary Hooper's persistence won Celtic a 68th minute penalty after he was tripped by David Proctor.
Scott Brown scored for the third game in a row as he expertly dispatched the spot-kick as Neil Lennon's side, who sit top of the Scottish Premier League and have already clinched a place in the League Cup final, made it 15 wins on the trot.
"It was a great performance and it could have been three of four in the end," the Celtic manager said of the cup holders.
"I thought we were superb from start to finish and I couldn't be more happy with my players today."
At the Caledonian Staidum, Celtic made four changes with Kris Commons starting his first match since November while Inverness brought in Tuffey for regular keeper Ryan Esson.
Hooper had an early chance for Celtic when he chased down a miscued clearance from Inverness but his backheel effort was blocked by an Inverness defender.
Both sides were struggling in the blustery conditions but Inverness came close to opening the scoring in the 24th minute.
Ross got on the end of Gregory Tade's headed knock down but Fraser Forster superbly tipped his low drive from 18 yards round the post before Kelvin Wilson stopped Meekings header on the line from the resultant corner.
It was then Celtic's turn to go on the attack with Commons completely failing to make contact with Hooper's cutback before the former saw his fierce shot on the turn held by Tuffey.
An Inverness mistake led to Celtic taking the lead in the 33rd minute.
Tokely failed to deal with Commons ball over the top from the halfway line allowing Samaras to break into the box and rifle a swerving shot off the outstretched hands of Tuffey and into the net.
Tokely was at fault again two minutes later when his soft back pass was intercepted by Hooper but Tuffey was quick off his line to stop his shot before showing good reactions to block Joe Ledley's follow-up effort.
The Inverness defender had a chance to make amends for his errors at the other end when he got on the end of Jonny Hayes' corner but he headed over the bar from six yards out.
Celtic began the second half on the front foot with Samaras spurning two chances to add to his tally when he ballooned Hooper's cross over the bar from six yards before he failed to connect properly with Ledley's ball in from the left wing.
Samaras sent another header straight to Tuffey as he met Charlie Mulgrew's free-kick before the Hoops were awarded a 68th minute penalty after Hooper was tripped by David Proctor in the box.
Brown stepped up to take the spot-kick and sent Tuffey the wrong way to slot low into the bottom left-hand corner to send Celtic into the draw for the quarter-finals.
"I'm bitterly disappointed and totally devastated as we've just not turned up today at all," Inverness boss Terry Butcher said.
"The first half an hour was pretty even but in the end it was total capitulation in the second half and I don't think Celtic will have had as easy a game as that up here."
Source: AFP
Source: AFP