The world number two won the match in Melbourne 6-3 6-7 (5-7) 6-2 6-3.
With Murray's compatriot Johanna Konta reaching the last four in the women's draw earlier today, Britain has two representatives in the semi-finals of a grand slam for the first time since 1977.
It is the sixth time Murray has advanced to the semi-finals of the tournament.
He was forced to scrap for almost every point but finally subdued his eighth seed rival after three hours and 20 minutes of gruelling baseline tennis in Rod Laver Arena.