He has also played in Nippon Professional Baseball (NPB) for the Tokyo Yakult Swallows and in the KBO League for the Samsung Lions.

On July 2, 2006, Suárez signed with the Tampa Bay Rays organization as an international free agent.

Suárez began his professional career in 2008, playing for the rookie ball Princeton Rays and went 0–2 with a 3.92 ERA in 11 games (nine starts).

He pitched for the GCL Rays and the Single-A Bowling Green Hot Rods in 2010, going a combined 2–5 with a 3.38 ERA in 15 games (14 starts).

[4] He was assigned to the Double-A Arkansas Travelers to begin the year, and after going 11–9 with a 2.98 ERA in 27 starts for the team, he earned a 2015 Texas League Mid-Season All-Star selection.

[7] Suárez made his major league debut on May 8, 2016, pitching one scoreless inning in relief against the Colorado Rockies.

[9] Suárez made his first major league start on June 1, 2016, against the Atlanta Braves, allowing 3 runs in 5 innings pitched.

[17] He started that day's game[18] and struck out the first batter he faced in the majors since September 26, 2017,[19] en route to 5+2⁄3 shutout innings.

[21] Suárez had a streak of three straight scoreless starts which ended when Jose Altuve hit a leadoff homer in the first inning of a 3–2 home victory over the Houston Astros on August 24.