McCann struck with a superbly executed left-footed free-kick in the 48th minute which sailed over the Diamonds' wall and into the top corner past Jamie Young, who had an inspired night himself.
Cheltenham's win gave them a six-point Christmas haul and means they have been beaten only once in their last 12 league and cup matches.
They deserved the points on the balance of play, although their own keeper Shane Higgs needed to be alert in the closing stages - notably when throwing himself to his left to claw away a great effort from Dave Savage.
The evening began on a sombre note with a minute's silence in memory of Diamonds secretary Dave Joyce, who died just before Christmas.
The opening exchanges were equally respectful.
Cheltenham carved out the best of the first half openings but McCann had a 20-yard shot which flew over the crossbar, while Brian Wilson was also to high with a 31st minute effort.
At the other end, Savage should have done better when he broke through but fired weakly, while on the stroke of half time substitute Chris O'Grady grazed a post after running and shooting from 35 yards out.
Cheltenham could have clinched victory straight after going ahead when Kayode Odejayi saw his shot rebound off Peter Hawkins, forcing Young to change direction and tip around the post. Then Steve Guinan broke clear only for the Diamonds keeper to save with his legs.
The introduction of 16-year-old sub Lee Tomlin almost salvaged a point for Rushden, who have now lost their last four home games. They must have known their luck was out when Tomlin's close range blast was blocked by a wall of yellow shirts.