Russell lost the first three points of the deciding tiebreaker before rallying to reach his second career ATP semi-final. He raised both arms in triumph when Sijsling hit a backhand return long and wide.
Russell and Sijsling took centre-court after a couple of rain delays pushed back play about two hours.
Russell won the final four points of the first tiebreak, but blew a 5-3 lead in the second set.
Nicolas Mahut, a wild card from France, faced Michal Przysiezny in another quarter-final after the lengthy opener.