At AaB he formed an attacking midfield duo with Peter Rasmussen, and especially the 1990 season was a breakthrough as Høgh scored 7 goals in 26 games.

He suffered a knee injury in late 1990, but Høgh managed to make his debut for the Denmark national team when he took part in the April 1991 friendly match 1–1 draw with Bulgaria.

He was one of several new players from Jutland clubs who arrived at Brøndby that year, including Marc Rieper who Høgh would later spend the majority of his Denmark national team matches partnering in the Danish central defense.

He also appeared in all Denmark's five matches at the 1998 FIFA World Cup, before he transferred to English club Chelsea for 3.5 million DKK in July 1999.

He won the 2000 FA Cup with the club, contributing two appearances to Chelsea's victorious campaign,[6] and he was also called up for Denmark to play the UEFA Euro 2000 tournament, though he sat out in all games through injury.