World number two McIlroy had to eagle the 72nd hole just to force a play-off after Lagergren's own remarkable eagle at the 16th at the K Club near Dublin.
After the first two additional holes were tied in birdie fours, Sweden's Lagergren found the water third time around to allow McIlroy to win with two putts.
"I just feel so lucky that I get to do this in front of these people," said McIlroy. "The support has been amazing all week. So happy I could play the way I did this week and get the win."
Victory gave Northern Ireland's McIlroy his second Irish Open title nine years after his first.
The 36-year-old became just the sixth man to complete a career Grand Slam of major titles following his Masters triumph at Augusta National.
McIlroy is warming up for the Ryder Cup at Bethpage Black later this month, when he will be leading Europe's defence of the trophy they won against the United States in Rome two years ago.