First-half goals from Ledley and James Forrest eased the anxiousness which had gripped the home side for much of the opening period.
Striker Anthony Stokes missed a great chance after the break and as the enthusiasm of Inverness was doused, Celtic cruised through the second half to stay on the tails of their Old Firm rivals.
Neil Lennon's men can look forward to Thursday's Europa League clash with Udinese at Celtic Park with confidence, but the Hoops manager will be aware the Italians present an altogether different challenge to the SPL's bottom side.
Manager Lennon was forced to make two changes to the side which beat Ross County in the Scottish Communities League Cup in midweek.
Glenn Loovens replaced the injured Kelvin Wilson in defence and midfielder Ki Sung-yueng came in for the suspended Charlie Mulgrew.
Richie Foran passed a fitness test for the injury-hit Highlanders but manager Terry Butcher also made two changes, Aiden Chippendale and David Proctor coming in for Andrew Shinnie and Roman Golobart.
Inverness had deprived Celtic of five points last season, albeit with a vastly different squad, thus doing as much any side to prevent the Hoops wresting the title back from Rangers.
And there was no mistaking the nervousness inside the stadium in the early stages.
In the sixth minute Fraser Forster slipped as he came out to meet Gregory Tade who had raced on to a long ball from Greg Tansey but the Hoops goalkeeper blocked the shot from the striker and the ball ricocheted out for a goal kick.
Moments later, as Caley broke with purpose down the right, midfielder Nick Ross just failed to get on to the end of Foran's cross and the ball sped wide.
Celtic's first effort came in the 15th minute when striker Gary Hooper fired wide from long distance, moments before Loovens missed Ki's corner from the right when it looked like he might have got a header on goal.
Proctor picked up a head injury in trying to defend the corner and was replaced by Golobart and the disruption appeared to affect the visitors as soon afterwards, in the 28th minute, Celtic were ahead.
Ledley had plenty of space when he took a pass from stand-in skipper Beram Kayal and set himself before drilling a shot from 25 yards low past Inverness goalkeeper Ryan Esson and into his bottom right-hand corner.
Five minutes later the home side doubled their lead when Forrest raced through the middle on to a Ki pass to knock the ball past Esson from the edge of the box.
Caley stabilised themselves and three minutes from the break Tansey went close with a drive after working some space for himself inside the Celtic box, before a couple of Inverness corners flashed across the Hoops six-yard box.
Celtic should have started the second half in more composed fashion but still an anxiousness pervaded, especially when the Hoops defence was called into action.
In the 51st minute, though, the home side should have had another goal.
Forrest slipped the ball to Hooper inside the box and his cut-back was perfect for Anthony Stokes, but the Irishman blasted high over from eight yards.
A minute later Hooper was replaced by Mohamed Bangura and he appeared to quickly bring added energy to the Hoops attack as the Parkhead men took a grip of the game, although the powerful and strong-running Tade continued to be a thorn in the side of the Celtic defence.
There appeared little chance of an Inverness comeback although there was enough slack play in the Hoops defence at times to give them encouragement.
In the 75th minute Lennon replaced Ki with Georgios Samaras and Stokes with Paddy McCourt in an attempt to find some more goals, but five minutes later Foster made a great save from Foran's close-range diving header.
McCourt missed a good chance by heading wide from a Bangura cross but the game petered out on the pitch as the home fans eventually found their voices off it.