Yet, despite the disappointment of falling further than he anticipated, the Valley native's distress turned to elation when the hometown D-backs selected him in the third round with the 120th overall pick.
"If I had to fall, the Diamondbacks were the team I wanted to fall to," Barrett said. "Living in Arizona, I've gone to a bunch of Diamondbacks games. They were my first pick. I went to their predraft workout and did fairly well, I thought, so I was hoping I'd go to them."
Barrett recalled the seconds leading up the special moment when he saw the club's logo appear on the clock.
"I just remember waiting, and then I saw the D-backs' name came up," he said. "My family and I were all just sitting around the house, listening to the Draft and then all of the sudden the Diamondbacks pop up and we're all hoping they call my name. Then I heard my name, and we were all excited. I was extremely excited to hear it."
With Barrett's fastball peaking in the upper 90s and his 6-foot-4, 220-pound frame drawing comparisons to the likes of Heath Bell and Jonathan Broxton, the D-backs were more than pleased when they came on the clock with his name still available.
"We were certainly happy," D-backs scouting director Roy Montgomery said. "Anytime you can get a guy like Jake in the third round is good, then you add the fact that he's a local kid and we know him well. We're happy to have him and I think it's a win-win situation for the organization."
The D-backs had drafted seven Sun Devils prior to Tuesday, but never as high as they took Barrett. Only one former ASU player has suited up for the D-backs in the club's history, current shortstop Willie Bloomquist.
Barrett began his ASU career as a setup man as a freshman but transitioned to a starting role as a sophomore. The righty started 14 games for the Sun Devils in 2011, going 7-4 with a 4.14 ERA. He struck out 72 batters in 76 innings that year but a shoulder injury caused him to miss the final portion of the season.
This year as a junior, Barrett moved back to the bullpen as the Sun Devils' closer. His size and maximum-effort delivery suited him well for the relieving role. He recorded 11 saves and made 31 appearances with a career-low 1.62 ERA.
"I attacking hitters more now," Barrett said. "I'm throwing a lot more strikes and trusting my defense."
As for the D-backs' long-term plans for Barrett, the organization will wait and see whether to develop him as a starter or reliever.
"I don't want to pin him into any role now," Montgomery said. "But I will tell you we're comfortable with him in that role out of the bullpen. He seemed to be doing pretty well there."