Although Leon Knight pulled one back for the home side late on it was a disappointing day for Kenny Jackett's men.
The Swans gave debuts to Kevin Amankwaah, Thomas Butler and Darren Pratley but Lee Trundle and Alan Tate were both missing serving one-match suspension.
The Robins were without the injured Damien Spencer and Michael Townsend but Brian Wilson and Steve Gillespie had recovered for a place on the bench.
The first half was poor in quality and Cheltenham could have taken an 11th minute lead when a deep cross from John Melligan was met by McCann but his glancing header was way off target.
The Swans relied on Andy Robinson to force the pace and he was given a rousing reception by the home fans despite his contract dispute with the club.
A typical all-action display by the want-away midfielder almost brought about the opening goal but Robins keeper Shane Higgs positioned himself well to clutch one rasping drive.
The second half exploded into life within two minutes of the re-start when Cheltenham opened up a ragged home defence.
Robins skipper John Finnigan crossed high to the left and Odejayi chested the ball down and fired into the corner of the net.
Before the Swans could recover from that blow they found themselves two down.
With just 54 minutes gone Willy Gueret was slow to come out and gather leaving McCann the simple task of toe-poking into an empty net.
Jackett responded immediately by throwing on all three substitutes and withdrawing the ineffective Butler, Kristian O'Leary and Rory Fallon.
The new players helped raise the tempos with Adebayo Akinfenwa particularly causing the visitors problems with his strong runs into the danger area.
The home side clawed one back in the 75th minute when a run down the left-hand cross from substitute Kevin McLeod deluded Higgs in the visitors goal and Knight dived to head into the corner of the net.
Robinson and Akinfenwa went close as Swansea searched for the equaliser but the Robins held out for a shock victory.
For Swansea it was the same old story. They are good going forward but lacking in defensive stability and if they want to go one better than last season, Jackett will have to quickly address that problem.