The Bucs sent their third-round pick (No. 74) and one of their two fourth-round selections (No. 106) to the Kansas City Chiefs for the chance to take the Lou Groza Award winner.
Aguayo is the highest drafted kicker since 2005, when the New York Jets took Mike Nugent at No. 47 overall. He is the ninth kicker chosen in the first two rounds of the common draft era, which dates to 1967.
"Honestly, it was where I wanted to go," Aguayo said. "They had the pick at 74. I was just waiting it out, and all of a sudden, the call came in."
Aguayo produced a sterling college career. He made 69 of 78 field goals and all 198 extra points from 2013 to 2015. The most accurate kicker in NCAA history, Aguayo had a .9673 conversion rate (267-of-276). He set an ACC record for consecutive PATs made, with 198, and became the 12th NCAA kicker to never miss a PAT.
"The importance of special teams is paramount," Bucs general manager Jason Licht said. "I didn't want to risk it. I wanted to take him. I have a lot of confidence in him. I like the way that he's wired. I like the body of work that he has put out there, obviously. A great kicker can be the difference in several games."
In Tampa, Aguayo will be reunited with quarterback Jameis Winston, who spoke highly of the kicker when the two were teammates at Florida State in 2014.
"Robert is amazing," Winston said at the time. "I think he is our best offensive weapon. I don't think anybody else in the country has a kicker like him. We have the utmost confidence in his ability."
The Buccaneers, who took defensive end Noah Spence earlier in the second at No. 39, were able to trade up into the round because they gained the No. 106 selection from the Chicago Bears, who moved from No. 11 to No. 9 on Thursday.
"We had him targeted all along," Bucs coach Dirk Koetter said. "If you really think about what happened, we really just picked the kicker in the third round. The extra pick we got in the fourth, we gave it back up, used that same third-round pick we were going to use in the first place to take the kicker. That's exactly what happened."
The Bucs struggled with their kicking game early last season. They cut Kyle Brindza after he went 6-for-12 on field goal attempts and 6-for-8 on extra point tries in four games. Connor Barth made 23 of 28 field goals and 25 of 26 extra points in replacing Brindza.