The Portrush man, who earlier defeated Shane Lowry in an all-Irish tie, was beaten 1-up by the Australian on Saturday.
McDowell started in good form with two birdies on the first three holes, but it was on the back nine when inconsistency caught up with him.
Frustration grew following a string of loose tee shots and ragged play on the greens and Day took the lead for the first time at hole 11.
McDowell was back on terms at 14, but lost the next hole before rallying again at 16.
However a bogey-five at 17 ended the contest with a missed 10-footer from the sleeve of the green representing his last chance.
A two-putt from Day put him through to the semi-finals to face Matt Kuchar.