Australia's Thompson knocks Rune out of Queen's Club tennis
Thompson came from a set down to defeat Rune, a semi-finalist at the grass-court warm-up event for Wimbledon last year, 4-6, 7-6 (7/4), 6-3 in a match lasting more than two hours.
Meanwhile world number 10 Grigor Dimitrov, the 2014 Queen's champion, made short work of a 6-1, 6-2 win over France's Adrian Mannarino.
"I was very happy with the way I started the match," said Dimitrov. "I think I set the bar high and wanted to be steady the whole time.
"Serve and return I just tried to focus on, they are the fundamentals of this surface," added the Bulgarian, 33. "As the match went on I felt more solid and more secure, so I am very happy."
Frances Tiafoe of the United States had retired with a right hip injury in the third set of his opener against qualifier Rinky Hijikata, as the Australian went through to the round of 16.
Former Wimbledon runner-up Milos Raonic plays British number two Cam Norrie later on Monday.
Reigning Queen's and Wimbledon champion Carlos Alcaraz, fresh from winning the French Open, begins his defence of this ATP title against Argentina's Francisco Cerundolo on Tuesday, when British tennis great Andy Murray faces Alexei Popyrin.