Baird had been in contention all week in his 365th Tour event and looked likely to take the title as he opened up a two-shot lead on the final day.
But when the 41-year-old found the water on 18, Kirk's consistency paid dividends and a birdie on 17 proved vital as he took the crown by one shot from Baird and South African Tim Clark who had played himself into contention on Sunday.
The American duo had led coming into the final round and it was Baird who made the early running with birdies on two and four to move into the lead.
Kirk had started his day with six straight pars before birdies at seven and nine but it was Clark who was setting the pace with a blistering front-nine display.
The 37-year-old picked up a stroke on the second before going on a run of birdies from the fourth to the seventh to reach the turn in 30.
A further birdie at 10 put Clark within one shot of leader Baird before a bogey at 14 looked like it may derail his chances.
However, a birdie-birdie-birdie finish saw him post a round of 62 and the clubhouse lead at 13 under as the final pair came into their closing holes.
Kirk took the initiative as he picked up shots at 11 and 12 but a birdie from Baird on 13 got him back within one and he regained the lead on 14 with a further picked up shot as Kirk bogeyed.
Baird was presented with a golden chance to move further ahead on the par-five 15th with Kirk in a bunker but he three-putted for par and Kirk holed a 20-foot putt to register his own.
That swung the momentum to Kirk, who won the Viking Classic in 2011 for his only previous Tour victory, and he recorded a birdie on 17 to take both players up the 18th on 14 under.
And it was Baird who felt the pressure as he put his tee-shot into a fairway bunker and then found the water with his approach to end his hopes of victory.
It was a low-scoring day at Sea Island with Scott Brown, Brian Gay and John Senden all shooting well to finish on 12 under.
Brown finished with a 66 to record his joint-lowest round of the week while Gay and Australian Senden both fired 67s on Sunday.
Two shots further back was former US Open champion Webb Simpson alongside Matt Kuchar and Matt Every with all three players recording four-under par 66s to finish the week.
Sweden's Robert Karlsson was the highest-placed European in a group of six players two shots further back.