"I just want to win," Day told the official PGA Tour website.
"I wanted to win so bad that I felt with how I was playing, if I kept playing the way I was doing, I would be holding the trophy at the end of the week.
"That's what kept me going."
Day, who won last week's Arnold Palmer Invitational at Bay Hill, helped by texts from his friend Tiger Woods, still sidelined with long-term back problems, started strongly by rolling home a birdie on the opening hole to go one up.
But McIlroy, in fine form himself heading into the opening major at Augusta in 11 days' time, hit back with three birdies to lead one up through the turn.