Norrie currently sits 12th in the race to qualify for the season-ending tournament in Turin next month and needs a big few weeks to break into the top eight.
The British number one had beaten Danish teenager Rune three times in 2022 but came unstuck in their fourth meeting, with the 19-year-old fighting back from a set down to win 6-7 (4) 6-3 6-3 and reach the semi-finals.
Norrie has been playing catch-up since being forced into quarantine and ruled out of tournaments in Korea and Japan after testing positive for Covid-19.
He battled past Aslan Karatsev in his first match in the Swedish capital and recovered from losing a break in the opening set against Rune to take it on a tie-break.
But the teenager broke serve twice in the second set and then again in the eighth game of the decider to set up a semi-final against Australian Alex De Minaur.
Still alive in the tournament and looking well-placed for victory is top seed Stefanos Tsitsipas, who eased past Mikael Ymer 7-5, 6-3.
He will face Finland's Emil Ruusuvuori for a place in the final on Saturday.
In the quarter-finals of the European Open in Antwerp, British number two Dan Evans took the first set but was beaten in three by world number 10 Felix Auger-Aliassime.
Evans saved three break points in the second set to force a tie-break, but Canadian second seed Auger-Aliassime managed to take it to a decider.
Evans was eventually beaten 4-6 7-6 (4) 6-2 in two hours and 40 minutes.