Born in Odense, Høgh started playing football with local club Odense BK (OB). He made his senior debut in 1977 as an amateur (later to become semi-professional, and eventually full-time-professional), and was an understudy to Danish national team goalkeeper Mogens Therkildsen, as OB won the 1977 Danish championship. The following years he established himself as OB's starting goalkeeper, and debuted for the Danish under-21 national team in March 1979. He was a part of the OB team which won the 1982 Danish championship and 1983 Danish Cup, and was called up for the Danish national team by national team manager Sepp Piontek. He made his national team debut in May 1983, and was a part of the Danish team at the 1986 World Cup.