The Danish sixth seed came through 6-3, 4-6, 3-6, 6-4, 7-6(8) after Davidovich Fokina served underarm at 8/8 in the final-set tiebreaker.
Holger was quick to it, winning the point to go to match point, which he converted when the 31st seed netted a forehand.
"I didn't expect the serve like that but I was fast to reach it," said Rune after the four-hour clash, in which he fired 19 aces in his 61 winners.
Rune had saved match points in the 10th game of the final set and then recovered from 6-2 and 8-5 down in the breaker.
He will face either Frances Tiafoe or Grigor Dimitrov for a place in the quarter-finals.
For Davidovich Fokina, it was a second successive nightmare exit from the All England Club.
Twelve months ago, he was penalised for ball abuse at 7-9 down in the fifth-set breaker against Jiri Vesely and lost the match.
He received a point penalty, which gave Vesely victory.
