Goycochea was the substitute for Nery Pumpido both in River Plate and in the Argentine national team, and got his big break in the 1990 FIFA World Cup, when Pumpido was injured in Argentina's second group game against the Soviet Union.
Goycochea was then the goalkeeper in the penalty shootout in the quarterfinal game against the Yugoslavian team. In the semifinal game against Italy, he saved penalty kicks in the penalty shootout from players Roberto Donadoni and Aldo Serena. He was also close to saving the heavily criticized game-winning penalty kick of the 1990 World Cup final against Germany, which Argentina lost 0-1. He was chosen as the goalkeeper of the Cup's All-Star Team.
El Goyco also played a few months in 1991 for the team Stade Brestois 29 from Brest in the region of Brittany, which was then named Brest Armorique, and was in the French second division and included players David Ginola, Corentin Martins and Stéphane Guivarc'h. Nonetheless, the team went bankrupt in November of that year, and was moved to the third division and lost its professional-team status.
With the national team, Goycochea also won the 1993 Copa América. On the occasion of that 1993 Copa América tournament, Goycochea appeared on TV commercials for a Pepsi promotion by PepsiCo's division in Guayaquil. He also was on TV commercials for Adidas soccer apparel some years later.
His last name, Goycochea, spelled without e, but which other people of the same genealogy spell as Goycoechea, is derived from the Basque surname Goikoetxea meaning topmost house (from goiko "of the top" and etxe "house"). Sergio is frequently nicknamed thus El Vasco, but also El Goyco.
For Argentina, Goycochea played in the following competitions:
*1990 FIFA World Cup Italy (runners-up)
*Copa América 1991 Chile (champions)
*Copa América 1993 Ecuador (champions)
*1994 FIFA World Cup U.S. (second round, substitute)
Nowadays, Goycochea is a football journalist hosting Elegante Sport (Argentina's Canal 7) and he partnered Diego Maradona at La noche del Diez.