Velázquez has played in both the Mexican League and the winter Mexican Pacific League. In those leagues, his overall record through the 2017–18 winter season is 70–51 with an ERA of 3.85 and a WHIP of 1.340, with 846 strikeouts and 386 walks in innings pitched.
In February 2017, the Boston Red Sox purchased Velázquez's contract from Mexican League team Piratas de Campeche. During the 2017 season, he appeared in 19 games for the Triple-APawtucket Red Sox, all as starting pitcher. He had an 8–4 record with 2.21 ERA and 1.000 WHIP in 102 innings pitched. The Red Sox promoted Velázquez to the major leagues on May 18, 2017. He made his MLB debut that day, taking the loss against the Oakland Athletics after pitching five innings and giving up six runs on nine hits, while walking two and striking out four. He split time between Pawtucket and Boston through the end of the season. Overall, for the 2017 Boston Red Sox he appeared in eight games and compiled a 3–1 record with 2.92 ERA, 1.135 WHIP, 19 strikeouts, and 7 walks in innings pitched. Velázquez won the final spot in the starting rotation of the 2018 Red Sox under rookie manager Alex Cora, exiting spring training on the 25-man roster. He made two starts in April but otherwise served as a relief pitcher during April and May. On May 14, Velázquez was placed on the 10-day disabled list with a "low back strain." At the time, he had made ten appearances for the Red Sox with a 5–0 record and 2.10 ERA in innings pitched. He was sent on a rehabilitation assignment with Triple-A Pawtucket on May 19, and returned to the active roster on May 24. Overall for the 2018 season, Velázquez made 47 appearances, pitching 85 innings with a record of 7–2 with a 3.18 ERA. He was not included on Boston's postseason roster. Velázquez was included on Boston's Opening Day roster to start the 2019 season. On May 30, he was placed on the 10-day injured list with a low back strain. To that point in the season, Velázquez had a 1–3 record in 18 games with a 5.97 ERA. The team sent Velázquez on a rehabilitation assignment with the Lowell Spinners on June 15, and activated him on June 17. He returned to the injured list on June 19, again due to a low back sprain. He was activated from the injured list on June 29, optioned to Pawtucket on July 15, recalled to Boston on August 8, and returned to Pawtucket on August 11. Velázquez was recalled to Boston on September 1, when rosters expanded. Overall with the 2019 Red Sox, Velázquez appeared in 34 games, compiling a 1–4 record with 5.43 ERA and 49 strikeouts in innings. On March 5, 2020, Velázquez was designated for assignment in order to make room for the newly-signed Collin McHugh on the 40-man roster.
Baltimore Orioles
On March 8, 2020, the Baltimore Orioles claimed Velázquez off of waivers. He was outrighted off of the 40-man roster on July 24.
