A neat passing move on seven minutes saw Erik Skora feed Fuller on the edge of the box and the Jamaican international squared to Cresswell, who took one touch before firing a low drive underneath Gary Montgomery.
But the goal did not wake up McAllister's men and the Lilywhites could easily have added to their lead.
By the 16th minute Cresswell could have had a hat-trick - first when he latched on to a brilliant Paul McKenna throughball, but was forced wide by Calum Davenport, then when he got the better of his marker on the right touchline and cut inside, curling a long-range effort just wide of the far post.
Coventry's only real effort of the half came from Dean Gordon, whose thunderous left-foot drive from 25 yards was tipped on to the bar by David Lucas.
City's rookie keeper Montgomery was called into action again as the half drew to a close, full-back Graham Alexander ran on to a loose ball on the edge of the box and struck sweetly enough, but it was at the right height for Montgomery to tip over.
The home fans thought they had snatched a draw ten minutes from time when Davenport headed home from a McAllister corner.
But their hearts were broken with just two minutes of normal time remaining as Coventry failed to clear their lines properly and Eddie Lewis volleyed expertly under Montgomery.