The deciding moment came in the 1-0 success came in the 65th minute when Marley Watkins got away down the right side and fired in a cross which goalkeeper Lukasz Zaluska could only parry. The ball rebounded straight off Hoops defender Eoghan O'Connell, who was helpless as the ball rebounded into the net.
The goal gave the Caley Jags their first victory over the Glasgow giants on their own patch since May 2011.
Inverness retained the same line-up that beat Motherwell at Fir Park last week, with James Vincent missing out again with an ankle injury and Watkins keeping his place on the right side of midfield.
Celtic made 10 changes from the 1-1 draw with Maribor in midweek. Charlie Mulgrew was the only man to keep his place, with starts given to O'Connell, Dylan McGeouch, Liam Henderson and Filip Twardzik.
Greg Tansey had the first chance of the game after four minutes, with a free-kick which went around the wall and rippled the side-netting.
The former Stevenage man came close again in the 11th minute as Danny Williams' pass broke to him 25 yards from goal. Tansey struck the ball firmly but it bounced wide of Zaluska's near post.
Leigh Griffiths had Dean Brill worried with a shot into the side-netting, before the Inverness goalkeeper was forced to tip over Efe Ambrose's scooped effort.
The visitors were beginning to exert some pressure on Inverness and Brill's smart double save kept the game scoreless.
A slip by Josh Meekings allowed Teemu Pukki time and space to shoot and Brill blocked with his feet, before repelling the rebound for a corner.
Looking to continue their impressive home form, Inverness were denied a penalty in the 27th minute as Ryan Christie appeared to be brought down by Mulgrew in the area.
Referee Bobby Madden waved the appeal away and Celtic went down the other end and tested Brill again, this time through Kris Commons.
With 10 minutes to go until the break, Nir Biton rattled the crossbar for Celtic before Inverness came close through Gary Warren - Mulgrew back-pedalling in time to clear the defender's header off the line.
The first chance of the second half came for the hosts, with Zaluska nearly caught out of position from a corner. Christie's inswinger caused problems in the Celtic six-yard box and Josh Meekings got to it, but he was unable to get his header on target.
Other than that opportunity, the opening of the second half was largely stale - with the ball getting bogged down in midfield.
Griffiths shot straight at Brill on the break and Celtic were made to pay for their lack of incisiveness in the 65th minute through O'Connell's mistake which gave the hosts' the advantage.
Inverness were attacking at will now and could have doubled their lead with 15 minutes to go, when Christie curled a low effort just wide of Zaluska's goal.
The impressive Watkins had given Celtic's defence the runaround all day and with Inverness looking to hold on for the victory he was reshuffled to the centre-forward position with Billy McKay substituted.
A late breakaway nearly put the icing on the cake for the Caley Jags as Williams rolled in substitute Nick Ross - but Zaluska reacted well to parry his effort.