McCann headed home in the 36th minute after Michael Higdon had shot against the bar as the visitors appealed for offside.
Inverness also had a penalty claim rejected early in the second half but Falkirk held out comfortably to book their place in the last four.
Caley Thistle had looked the likelier team to make the breakthrough before McCann’s opener, although Graham Barrett was inches away from giving Falkirk a fourth-minute lead.
McCann’s cross found Barrett in space at the far post and he steered the ball over the onrushing Ryan Esson, but Grant Munro got back just in time to clear off the line.
Inverness created their first chance in the 13th minute when Dougie Imrie’s though ball released Don Cowie. The midfielder ran out of space after taking the ball round goalkeeper Scott Flinders, and shot into the side netting.
Falkirk, who brought in Higdon for the suspended Steve Lovell, enjoyed a decent spell but Munro made another crucial intervention when he dived to head behind Jackie McNamara’s dangerous cross.
Inverness then gained the edge and Flinders pushed clear Adam Rooney’s header after Richard Hastings had found the striker with a deep cross.
The Falkirk keeper then dived low to his left to push wide Jamie Duff’s first-time effort after a free-kick dropped for the defender around the penalty spot.
The visitors went close again when Falkirk failed to track another forward run by Cowie, who took Roy McBain’s pass in his stride and shot inches over from 20 yards.
The home side were struggling to pass the ball through Inverness but McNamara’s ball over the top handed them their disputed goal.
As most people inside the Falkirk Stadium awaited an offside flag, Higdon struck the bar from the edge of the area before McCann headed home the rebound with Esson stranded.
Inverness boss Craig Brewster was involved in a touchline row with referee David Somers and his assistant, and he had his head in his hands in the dying seconds when Ross Tokely missed the target with a close-range free header.
Hastings and Tokely were both booked for dissent, the latter just before the second half started, and the visitors had another grievance in the 55th minute.
Rooney fell under a challenge from Gerard Aafjes as he broke into the box but Somers waved play on. Aafjes did not appear to play the ball but Rooney’s team-mates, and Brewster, were more animated in their penalty appeals than the striker himself.
An unmarked Darren Barr almost killed off the contest just before the hour mark as he slid the ball off the post from 10 yards after McCann’s cross had caught out the Inverness defence following a half-cleared corner.
Barr was soon needed at the other end when Flinders dropped Imrie’s cross at the feet of Rooney, with the Falkirk skipper reacting quickly to block.
However, Inverness were finding chances harder to come by as Falkirk were far better in possession than they had been for much of the first half.
Esson, whose only two previous starts for Inverness came in this tournament, was not seriously troubled again though.
Rooney squandered a good chance to equalise with six minutes left as he shot weakly at Flinders from 10 yards following Cowie’s cross from the left.
McCann’s shot trickled wide at the end of a long run forward and Flinders saw out the game in relative comfort, although he was fouled by his opposite number as Inverness threw everyone forward for an injury-time corner.