Derrick Williams gave the League One leaders a 30th-minute lead, rising to head home Luke Freeman’s free-kick from the right after Reda Johnson had been cautioned for bringing down Mark Little.
And Wilbraham made sure of a two-legged final against Gillingham by powerfully heading in another Freeman cross 15 minutes from time.
Coventry had a strong penalty claim on 34 minutes when Luke Ayling went to ground to challenge Gary Madine inside the box, but referee Stephen Martin ruled the defender had got a touch on the ball.
Madine shot straight at Frank Fielding from a good early chance and Johnson headed into the side-netting as the visitors failed to make the most of some promising approach play.
The hosts defended strongly throughout and are still involved in two cup competitions as well as on course for promotion.