Garcia, 34, carded a six under final-round 66 to tie for second with American Rickie Fowler on 15 under.
The Spaniard lost the 2007 Open in a play-off and was also runner-up at the US PGA Championship in 1999 and 2008.
"I felt I did almost everything I could and there was a better player here - it's as simple as that," he said.
"I'm going to keep trying, I'm going to keep putting myself in these situations."
'Lucky' ricochet for Sergio on 12thGarcia, who trailed McIlroy by seven strokes at the start of the final round, had a moment of outrageous luck at the 12th hole when his ball hit the grandstand but ended up near the green.
But he was left to rue an error at the par-three 15th where he first hit his tee shot into a bunker before he failed to escape the greenside trap with his first attempt.
"Obviously 15 was a mistake but I was trying as hard as I could and I tried to get too cute," said Garcia, who has won 19 times on the PGA and European Tours.
"I wanted to make him (McIlroy) feel a bit of pressure and see how he would respond - he obviously responded well."
Garcia said he was overwhelmed by the support he received from the crowd, especially on the final hole.
"With that cheer on 18 I thought I was going to faint," added Garcia. "That's one of the reasons I love this championship so much. To win it would definitely make my career.
"I'm not disappointed. I played hard and gave it everything."
Sergio Garcia (left) and Rickie Fowler finished two shots behind Rory McIlroy at the 2014 Open Fowler, 25, has now finished second twice and fifth in 2014's first three majors. He said: "It's hard to be disappointed, because it was such a great week.
"It's similar to being one of the only guys at the US Open to be under par and not win.
"There's plenty more to come. I'll take 15-under in a lot of majors and sit there and wait in the clubhouse."
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