Chris Greenwood finished alone in third at 15-under 265 while Hank Kuehne was one shot further back at 14-under 266.
Walker began the day in a tie for the lead alongside Ken Duke. Walker got off to a shaky start with a bogey at the opening hole but made up for it with an eagle at the par-five fourth.
He birdied the 10th before running into trouble with a bogey at the following hole. The 23-year-old remained calm and ran home back-to-back birdies from the 13th to climb to 18-under.
Gillespie provided the challenge with an eagle of his own at the 10th and birdies over four of the remaining seven holes.
At the 18th, Walker hit a brilliant approach shot and ran home the birdie putt to secure the victory.
'You look at the scores that have won tournaments out here, I just decided to come in here this week and try to make as many birdies as I can,' said Walker. 'I figured Derek may have made one coming in, and then I looked at the leaderboard and knew I had to hit it close. But I didn't want to be smiling on the fairway. I just wanted to get my business done with the putt.'
Walker made the cut at last year's U.S. Open. He finished tied for 47th at the first leg of the Scottsdale Swing last week at McCormick Ranch.
Duke struggled to a 73 that included a triple bogey at the 16th. To make things worse, an additional two strokes were added following the completion of the round when it was discovered that his ball moved during a practice swing.
He finished at 12-under 268 alongside Chris Wisler and Rafael Gemoets.
Full-field scores from the Scottsdale Swing at Eagle Mountain