Angelini attended Junípero Serra High School in San Mateo, California, where he excelled in baseball.

Angelini attended the College of San Mateo from 1965 to 1968, where baseball head coach John Noce developed him into a good prospect.

Angelini instead decided to attend Washington State University, where he played college baseball for the Cougars from 1968 to 1969.

On May 22, 1969, while facing the Idaho Vandals, Angelini finished his college pitching career by striking out 18 batters.

Before the 1969 MLB season, Angelini was signed by the Kansas City Royals as an amateur free agent.

He struck out 74 batters in 72 innings pitched and was promoted to the A-ball San Jose Bees of the California League in 1970.

On July 22, 1972, at Municipal Stadium, in the top of the seventh inning and trailing 3–2 to the Baltimore Orioles, Angelini made his Major League debut, relieving Mike Hedlund and appearing as the third Royals pitcher in the ballgame.

Angelini gave up a two-run home run to first baseman Boog Powell on the first pitch and strike he threw in the Majors.

“But then I struck out Davey Johnson on four pitches and it pumped me up for the next inning.” Angelini pitched a scoreless eighth inning and the Royals proceeded to rally for four runs in the bottom of the eighth and ultimately won the game 8–5, making Angelini the winning pitcher in his Major League debut.

On August 14 at Municipal Stadium, in a 3–0 win over the New York Yankees, Angelini earned his first career save after getting the last 2 outs to finish the ballgame.

On June 19, 1973, at Royals Stadium, pitching against the Oakland Athletics, Angelini would appear in his final Major League game.