Victor Wanyama put the champions ahead after two minutes when he found space from an Adam Matthews corner to head home powerfully.
A Charlie Mulgrew free kick rattled the crossbar just moments later as Celtic threatened to run away with the match in the first half.
Watt scored the first of his goals just before the break as he accelerated past the Inverness defence before finishing well past Ryan Esson.
Celtic continued to dominate the match in the second half but an improved Inverness side frustrated Neil Lennon's attacking players.
However, Mulgrew scored Celtic's third as he fired a low shot past a helpless Esson from outside the area.
Watt rounded off an impressive performance with his second of the match as he took advantage of some uncertainty inside the area to slot the ball home.
However, some poor defending from Celtic late on took some shine off the victory as Ross Draper fired past Fraser Forster after running onto a header.
Draper scored again with only a few minutes remaining as he reacted to a Forster save to finish easily from close range.