Home keeper Ben Purdham then made two splendid saves to deny Will Whieldon and then Izak Reid.
The hosts began to take control after the half hour and Curtis Pond made a number of good saves, the best being from Antoine Recizac who had just the keeper to beat.
Chasetown upped the pressure at the start of the second half and took the lead in the 63rd. After Chasetown’s eleventh corner, Tom Hill laid the ball back to Jack Langston who drilled it low past Purdham.
Clitheroe were a whisker away from the equaliser. They finally got the ball beyond Pond but Harry Pratt saw his effort headed off the line by James Askey.
Chasetown punished their hosts with a superb second goal . Left back Callum Cockerill-Mollett received the ball from substitute George Cater into the box and smashed the ball past Purdham at his near post.
In stoppage time, Hill notched his second assist with a through ball to Danny Cocks who slid the ball confidently home.