The Northern Irishman had six birdies and two bogeys in his round to set the clubhouse lead in Ohio on 13 under.
But Americans Jon Curran (70) and McGirt (71) finished on 15 under, the latter after holing a four-foot putt to force a play-off at Muirfield Village.
McGirt won with a par at the second extra hole after both players missed the green with their approach shots.
Curran, who missed a 30-foot putt to win his first PGA Tour title on the first extra hole, found a fairway bunker with his tee shot when they replayed the 18th, flew the green with his second and floated his third 30 feet past the pin before two-putting for bogey.
McGirt, who had also found the greenside rough with his second shot, chipped to seven feet and holed the uphill putt for par to record his first PGA Tour win.
"It was a struggle all day but to get your first win is pretty special," said the 35-year-old, who had been in a three-way tie for the lead overnight with compatriots Matt Kuchar and Gary Woodland.
Fellow American Dustin Johnson, who was 16 under at the turn, had a two-over-par back nine to finish third on 14 under.
