The pair are vying for glory at Weymouth and Portland, with Hogh-Christensen leading the way, thanks in no small part to finishing ahead of his British rival for the first six races.
Ainslie dramatically boosted his chances of gold on Day 6 by narrowing his rival's lead from 10 points to three, although it was the latter who came off the better in race nine this afternoon.
Hogh-Christensen got off to by far the better start and rounded the top mark fourth in the fleet.
Three-time gold medallist Ainslie languished 11th but managed to move up to eighth around the next mark, although 100m separated him and the Dane.
Ainslie continued to break out of the pack as racing went on and he rounded the fifth mark in ninth position, although that did not stop him chasing down Hogh-Christensen.
Only one second separated the pair around the final mark, although it was the class leader who held out to finish fifth ahead of Ainslie, thus extending his lead at the top of the standings to four points.
The Netherlands' Pieter-Jan Postma won the race by 31 seconds.