Kelliher had Burns, at 16/1 with Planet Sport Bet before the tournament began, as his headline selection believing him to be in the perfect position to take advantage of a weak field at the Country Club of Jackson.
Kelliher said before the tournament: "Burns was third here in 2018, is a player that excels on the Bermuda grass putting surfaces and one of the top five putters in the field statistically and an elite level iron player.
"He is a player that can win many tournaments throughout his career. He has a great chance to take advantage of a relatively weak field and get another PGA Tour win under his belt."
Burns picked up four shots on the back nine has he collected a 67 to win the Sanderson Farms Championship.
It was the second PGA Tour title in 2021 for the American after victory in May's Valspar Championship, with the 25-year-old jumping one spot on the leaderboard after seven birdies in his final round in Jackson.
He bogeyed the eighth and the 18th, but his two-shot lead going into the last was enough of a cushion as he finished on 22 under.
Nick Watney impressed on Sunday with his 65 moving him into a tie for second alongside Cameron Young on 21 under.
A trio of Americans - Hayden Buckley, Andrew Landry and Trey Millinax - tied for fourth on 20 under along with Sweden's Henrik Norlander, while overnight leader Sahith Theegala shot a 71 and fell to a tie for eighth.