The win was the Argentine Reutemann's tenth Formula One victory and his first since the 1978 United States Grand Prix.
Piquet's third place moved him past René Arnoux and Alan Jones into the lead of the world championship for the first time.
The race however is remembered for a memorable and spectacular crash at the start of the race when Derek Daly collided with Bruno Giacomelli's Alfa Romeo 179, which sent Daly's Tyrrell 010 flying over Giacomelli and landing between teammate Jean-Pierre Jarier and Alain Prost's McLaren M29.
The accident was particularly disappointing for McLaren after John Watson failed to qualify for Monaco's shortened 20 car grid.
Fourth was West German driver Jochen Mass driving an Arrows A3 one lap down on Reutemann ahead of Canadian Gilles Villeneuve in a Ferrari 312T5.