The tournament was delayed by rain and both matches were played at the same time in order to complete them before nightfall.
Cilic broke Hewitt twice in the third set to advance. Hewitt is a four-time Queens Club champion.
Murray, who defeated Tsonga in the 2011 Queens Club final, broke serve twice in the third set.
The Scotsman appeared to have the advantage over Cilic heading into Sunday's final. Murray is 8-1 lifetime against Cilic.