Angotti was traded twice June 11, 1968, first to the Blues by the Flyers and then to the Pittsburgh Penguins. He was then traded to the Blues by the Penguins on June 6, 1969 and claimed by the Black Hawks in the NHL Intraleague Draft five days later. He played the next four seasons for Chicago, which lost the 1971 and 1973 Cup Final to the Montreal Canadiens in seven and six games, respectively, and was a player-coach for the Blues after Jean-Guy Talbot was fired with 23 games remaining in the 1973-74 season.
Angotti was fired nine games into the 1974-75 season (2-5-2) and went to the World Hockey Association to play 26 games for the Chicago Cougars. He returned to the NHL as Penguins coach in 1983-84 (16-58-6).
The Toronto native is one of three Black Hawks with a four-point game in the Stanley Cup Final (Stan Mikita, Game 5, 1973; Dustin Byfuglien, Game 5, 2010), scoring two goals with two assists in a 5-3 victory in Game 2 of the '71 Final. He scored 289 points (103 goals, 186 assists) in 653 regular-season games and 16 points (eight goals, eight assists) in 65 postseason games.
Angotti won the 1962 NCAA championship with Michigan Tech University, two years after losing it, and was named MVP of each tournament. He scored 17 points (nine goals, eight assists) playing an NHL-high 70 games for the New York Rangers in 1964-65 and was traded to the Black Hawks for cash 21 games into the 1965-66 season.