Taylor was a first-team All-Big 12 selection last season when he had 13 tackles for loss and seven sacks, and also tied for the FBS lead with four blocked kicks.
Godchaux entered the draft as an underclassman after spending three seasons at LSU and had 7.5 tackles and 5.5 sacks last season.
Asiata was a second-team All-Pac-12 selection in 2016 after receiving honorable mention accolades the previous season.
The Dolphins added another pair of defensive players in Day 2, getting a physical linebacker and a productive cornerback, both selections making plenty of sense.
The selections of defensive end Charles Harris, linebacker Raekwon McMillan and cornerback Cordrea Tankersley mark the first time in team history the Dolphins have used their top three draft picks on defensive players.
Tankersley and Byron Maxwell never played together at Clemson, but the newest Dolphin said the two have a close relationship.
Tankersley had nine interceptions over the past two seasons, including four last season when Clemson won the national title.
McMillan playing in 14 games for the national champion Buckeyes as a true freshman in 2014.