Manager Ronny Delia made 10 alterations to the team that drew 1-1 at Maribor in Wednesday’s UEFA Champions League play-off first leg.
Nir Biton curled a shot against the crossbar in the first half, but the game's decisive moment arrived in the 65th minute when Eoghan O'Connell helplessly put through his own goal after Marley Watkins' cross was parried in his direction by Celtic goalkeeper Lukasz Zaluska.
The result means Inverness remain top of the fledgling Premiership table with 10 points from four games, one ahead of Hamilton Academical and Dundee United - both of whom were respective 2-1 winners over Partick Thistle and Ross County.
Dundee United did not endure quite such a breathless finale, but required an emphatic Chris Erskine finish four minutes from time to seal the points.
Nadir Ciftci gave the Tannadice faithful something to cheer in the 19th minute as he opened the scoring from Paul Paton's free-kick, but Jake Jervis steered home to bring County level six minutes into the second half.
Youngster Robbie Muirhead rifled a magnificent opener from 25 yards and Motherwell's Fraser Kerr saw red for a high challenge on Jamie Hamill, with Sammy Clingan netting the resulting free kick.
