Murray, 29, is the first Briton through to a final in a successful tournament for British tennis.
Johanna Konta lost in the women's singles semi-finals, while Andy Murray, Jamie's younger brother, plays Milos Raonic in the men's last four on Friday.
"Somebody must be doing something right over in the UK," said Jamie Murray.
"It's been great. Andy's obviously been doing amazing things for a long time and what Jo's done this week has been quite unbelievable.