Leggatt is the third straight first-time winner at Tucson and the 11th in the tournament's 56 years.
The 36-year-old Canadian, who attended Q-school last year after just failing to keep his card as a top 125 money-winner in his rookie season, tied his career-low with a final-round 8-under-par 64.
He had a 72-hole total of 20-under 268, the lowest score at Tucson since David Frost's 266 in 1988.
Roberts fired a 66, but hurt his chances with a late bogey and four makeable putts in the last six holes that wouldn't drop. Peoples shot 67.
Kenneth Staton and Fred Funk came in at 271, Staton after a 64 and Funk after a 68.
Spike McRoy (64), Cameron Beckman (67), Russ Cochran (67), Chris Smith (68), Shigeki Maruyama (68), Bob Tway (69) and Greg Kraft (69) were grouped at 272.
Heath Slocum, who had the lead after shooting 64 on Saturday, struggled to a 72 and a 272 total. Andrew Magee (74) and Brandel Chamblee (75), other members of the last threesome, struggled in the final round and finished at 275 and 276, respectively.
The $540,000 winner's purse is more than Leggatt's combined earnings since turning pro in 1990.
He entered two events each in 1999 and 2000, but made 12 cuts in 29 starts as a rookie last year and earned $368,862 ' finishing 133rd overall. He arrived in Tucson 79th on the 2002 money list after making three cuts.
This time, Leggatt played with purpose, an uncanny short game and a champion's nerve to catch and pass the 15 players who began the round with equal or better scores.
He faced a tricky situation on the 18th hole, a 465-yard, par-4, when his approach shot bounced over the green and into a bunker above the flag. Leggatt's next shot stopped 10 inches from the cup.
It was his second straight brilliant shot from around green.
On the 17th hole, Leggatt chipped to within 5 feet and sank the birdie putt to open his two-shot lead.
He carded a tournament-best 6-under on the front nine to reach 18-under, also birdied No. 10 and then gave back a shot with a bogey on No. 13.
Roberts, playing a hole ahead, made four straight birdies in a surge ending at No. 11 to get within a shot, and shared the lead after Leggatt's bogey.
But Leggatt chipped to within 2 feet on the par-5 15th hole and made that putt to get back to 19-under.
Full-field scores from the Touchstone Energy Tucson Open