The Australian produced a stunning eagle from 50 feet on the 16th to move into the lead for the first time, and he held his nerve with back-to-back pars on the final two holes to take an 11-under lead into the clubhouse.
Overnight leaders Kisner and Hoffman were unable to chase him down, with the former failing to find the birdie he needed to force a playoff and Hoffman recovering from a bogey on the 17th with a birdie on the last to ensure a tie of second place, one behind Leishman.
McIlroy also forced his way into the equation despite having been four-over-par at one stage earlier in the tournament, powering back from a five-shot deficit at the start of play on the final day.
However, the Northern Irishman, who will still climb up to second in the world rankings, blew his chance on the final hole after seeing Leishman take control with his eagle and was forced to settle for third place alongside England's Tyrrell Hatton.
"I thought going out today if I could shoot anything in the 60s it would be a good score. Obviously I was trying to shoot a couple lower than that but overall played well, I'm really pleased with how I went this weekend and can take a lot of positives from it," McIlroy told Sky Sports News.
"I saw Leishman eagle 16 so I thought I really needed to try and hole it and I was overly aggressive with it and missed the one coming back, but it's been a good week and if anything, to be one over after two rounds and come up like this, it's a great weekend.
"I can't complain too much. It would have been nice to give it a bit of a better go, I had two chances on 17 and 18 there, but hopefully I can get back up on the horse next week and try and get a win at the (WGC-Dell Technologies) Match Play."
Leishman added: "It's amazing. It's been quite a while and my son keeps reminding me - daddy why don't you ever win the trophy? I've been trying extra hard for him and it was great today that he was here and a special day with everything honouring Mr Palmer.
"You always want to win to honour the greats and it's a special year this year. It would have been great if he was up there standing on the back of the green but to win at his place is a dream come true."
The victory is only Leishman's second PGA Tour title.