Until he drove home a downward header from Louis Carey with seven minutes to go a rare goal from midfielder David Bell had looked likely to bring the Sky Blues their first win over the Robins for 30 years.
They had seen their goal peppered late on by anxious shots from a side beaten 6-0 by Cardiff in their last home match, but looked comfortable repelling everything until Clarkson scored ten minutes after coming on.
It was always a game looking for a hint of inspiration and the best chances were Bristol's in the first half.
Coventry's captain Stephen Wright provided the first of them when he bumbled into Patrick Agyemang a yard outside the box. It was a perfect place for dead-ball specialist Paul Hartley, but he could only lift his kick into the wall.
Bristol pressing without threatening, but with half-time approaching central defender Carey found Nicky Maynard in the penalty box with a nicely waited 50-yard ball.
An opening goal was a shot away but Maynard was denied that by a superbly timed tackle from James Cake, which left him on his back and the ball rolling away towards the touchline.
Coventry having sold their top scorer Leon Bent to Newcastle on transfer deadline day relied on Michael McIndoe, a Robins player until last summer, to create something for them with his speed on the left.
He occasionally got the better of Bradley Orr without the final worrying ball.
The breakthrough for the visitors came 53 seconds into the second half. Sammy Clingan's shot bounced back off a defender to Bell, whose opportunist strike from 25 yards caught the keeper on the wrong side of his goal.