The home side deservedly took the lead after a sustained period of pressure.
A Sam Luke throw in was flicked on by a defender and Jake Lovatt netted his fifth goal of the season with a crisp left volley. Clitheroe levelled in the 36th minute.
Josh Gregory crossed for Andreas Bianga who drilled low into the bottom corner.
Levi Rowley was then a whisker wide with a header as Chasetown tried to regain the initiative. Both sides created chances early in the second half.
Curtis Pond saved a free kick from Clitheroe’s Jonah Gosling while Chase skipper Jack Langston shot on the turn no further than a foot wide of the upright.
No side could force the winner though and honours were shared.