serving at deuce, Rune hit an overhead winner to reach gamepoint.
To end an extended rally, Djokovic struck a down the line backhand long to drop serve and trail 0-3.
Rune struck a crosscourt forehand passing shot winner to lead 15-0 before Djokovic hit several groundstroke errors to trail 0-4.
Djokovic struck an overhead winner and an ace out wide in the following game to hold serve to reach 4-1.
Djokovic committed two forehand errors before Rune hit an inside out crosscourt forehand winner to hold serve and lead 5-1.
Djokovic struck an overhead winner before Rune committed two crosscourt backhand errors to drop the following game and reach 5-2.
With Rune serving for the match in the final game of the match, Djokovic reached breakpoint at 30-40 before hitting a down the line backhand long to reach deuce.
Djokovic struck a crosscourt forehand long before also hitting an inside out down the line forehand long to end the match in his 35th unforced error for the day.
Following the hard fought contest, the two players hugged at the net.
Rune takes on the 4th seeded Casper Ruud of Norway in Saturday's semifinals.
Should Rune win his semifinal, on Monday he would move from seven to six in the professional tennis rankings.
If the world number five Stefanos Tsitsipas of Greece loses his quarterfinal match later today and Rune wins the Rome tournament, Rune would move from number seven to number five in the tennis rankings on Monday.