He is known for being the first (and thus far, the only) San Diego Padres pitcher to win a World Series game. Hawkins earned a victory pitching in relief in Game 2 of the 1984 World Series, which the Padres lost to Detroit in five games. His best season was 1985, when he threw a career-high 2282⁄3 innings, compiled an 18–8 record, (winning his first 10 starts) and finished with a 3.15 ERA. Hawkins is the only pitcher to win his first 10 starts since the advent of divisional play in Major League Baseball which started in 1969. In 3 career starts at Fenway Park, Hawkins managed a total of just one inning, giving up 18 earned runs in those starts.
On July 1, 1990, Hawkins pitched a no-hitter for the Yankees against the Chicago White Sox at Comiskey Park but lost the game. Hawkins dominated the White Sox into the eighth inning, but after retiring the first two batters, Sammy Sosa reached on a fielding error by Yankees third baseman Mike Blowers. After Hawkins loaded the bases by walking the next two batters, Robin Ventura lofted a fly ball to left field. Rookie Jim Leyritz, fighting a blustery wind, had the ball glance off his glove for an error, allowing all three baserunners to score. The next batter, Ivan Calderón, hit a fly ball to right field, which Jesse Barfield lost in the sun and dropped for another error, allowing Ventura to score. The Yankees could not score in the ninth, giving Hawkins the loss despite not allowing a hit.
The 4–0 loss was the largest margin of a no-hitter loss in the 20th century, and Hawkins became the first Yankees pitcher to lose a no-hitter. On September 4, 1991 the Committee for Statistical Accuracy, appointed by Commissioner Fay Vincent, changed the definition of a no-hitter to require that a pitcher throw at least nine full innings and a complete game for the no-hitter to be official. Since Hawkins played for the visiting team, the White Sox never batted in the ninth inning and Hawkins lost the credit for a no-hitter.
A high-priced free agent in the second year of a three-year deal, Hawkins struggled for a poor Yankees team in 1990. On May 8, with just one win and an ERA of 8.56, Hawkins was offered his outright release, which he accepted. But an injury that night to pitcher Mike Witt changed his mind. Hawkins pitched much better in his next three starts, although he still had only a 1–4 record prior to the no-hitter.
In his next appearance, he faced the Minnesota Twins at Yankee Stadium in the opening game of a doubleheader. Hawkins pitched a shutout into the twelfth inning but wound up losing 2–0 – the last time a starting pitcher has pitched into the twelfth inning or later. In his following appearance, the Yankees lost a six-inning no-hitter to Mélido Pérez and the White Sox.