Bublik and Murray have now beaten each other two times apiece this year after the No.3 seed won 7-5 6-4 in Newport.
The Kazakh overpowered the three-time grand slam champion to move into the last four, winning 80 per cent of his first-serve points and breaking three times.
Murray hit back straight after being broken at the start of the second set, but the former world No.1 trailed 4-3 after failing to hold for a third time and Bublik did not give him another look-in.
Bublik will do battle with Jason Kubler for a place in the final after he defeated fellow Australian James Duckworth 7-5 7-6 (7-3).
Sebastian Baez ended Thiem's encouraging run to the quarter-finals with a 6-2 6-7 (5-7) 6-4 win in Bastad.
Thiem, who has endured an injury nightmare, won his first match 14 months by beating Emil Ruusuvuori in the first round and followed that up by getting the better of Roberto Bautista Agut, but the former US Open champion will go no further.
Argentinian Baez will now come up against Andrey Rublev after the No.2 seed from Russia saw off Laslo Djere 7-5 6-4.
Pablo Carreno Busta blew Diego Schwartzman away in a 6-1 6-0 thrashing, while Francisco Cerundolo got past Aslan Karatsev in straight sets.