[5] 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.

Overall, for the 2017 Boston Red Sox he appeared in eight games (three starts) and compiled a 3–1 record with 2.92 ERA, 1.135 WHIP, 19 strikeouts, and 7 walks in 24+2⁄3 innings pitched.

Velázquez won the final spot in the starting rotation of the 2018 Red Sox under rookie manager Alex Cora,[7] exiting spring training on the 25-man roster.

"[8] At the time, he had made ten appearances for the Red Sox (two starts) with a 5–0 record and 2.10 ERA in 25+2⁄3 innings pitched.

[16] Overall with the 2019 Red Sox, Velázquez appeared in 34 games (eight starts), compiling a 1–4 record with 5.43 ERA and 49 strikeouts in 56+1⁄3 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.

[23] On July 14, 2021, Velázquez signed a contract to return to the Acereros de Monclova of the Mexican League, with whom he played for in 2016.

[26] On April 9, 2024, Velázquez signed with El Águila de Veracruz of the Mexican League.

[28] On October 11, 2024, Velázquez was hired as the pitching coach for the Rieleros de Aguascalientes of the Mexican League.

[29] Velázquez has played in the Mexican Pacific League since 2010, and represented Mexico at the 2017 and 2021 editions of the Caribbean Series.

[30] Velázquez was on the Mexico national baseball team roster for the 2020 Summer Olympics (contested in July 2021), but a positive COVID-19 result disqualified him.