The Spaniard led by two shots at the start of the fourth round but finished with a two-over 74 to join Kim, who shot a 71, on nine under for the tournament.
Both players required three putts on the 18th on the first hole of the play-off, but after teeing off again on the 18th it was Recari who secured a birdie, and the win, with an 18-foot putt.
"There was a lot going on," Recari said. "I had a good feeling, but the same as the other putts.
"The other ones went by and missed the hole. This one went in. I had the same process and the same shot. Nothing different.
"I'm always trying to do my best. It just dropped in.
"Part of it I still don't remember. Just a lot of emotions. It's a different win; it's my first in a playoff.
"It's a lot of different things, different situation. I'm definitely more emotional than my first and I don't know what my third will bring."