He has Austrian roots on his father's side, since the city of Udine was part of the Austro-Hungarian Empire before WW1.
Due to his imposing stature and physique, as well as his tenacious style of play, Inter team-mate Armando Picchi (who was the captain and sweeper of the side) gave him the nickname "La Roccia" (The Rock).
A strong, large, quick, and energetic player, Burgnich is regarded as one of the greatest Italian defenders of his time. He was a versatile player who was capable of aiding his team both offensively and defensively, and he was capable of playing as a centre-back, as a sweeper, and as a right-sided full-back, where he particularly excelled, forming a formidable partnership with Giacinto Facchetti at Inter and with the Italian national side. He was an excellent man-marker and tackler, who was adept at anticipating his opponents; he is also remembered for his discipline and for his concentration. He played 467 times for the Nerazzurri, scoring 6 goals, and his physical, aggressive, determined style was ideally suited to the catenaccio system operated by Helenio Herrera throughout Inter's glory years. With Inter, Burgnich enjoyed a highly successful period of domestic, European, and international dominance, winning five Italian championships, two European Cups and two Intercontinental Cups. He was notably part of the legendary Inter lineup of the 1960s still known today as the Grande Inter.
Following his 12 seasons with Inter, he transferred to Napoli in 1974 for the final three seasons of his career, finally winning the Coppa Italia, as well as the Anglo-Italian League Cup in 1976, before retiring in 1977. In total, he made 494 appearances in Serie A throughout his career.
He may best be membered for his quote about Brazilian star Pelé's headed goal against him, following Italy's 4–1 defeat to Brazil in the 1970 World Cup Final (Burgnich had been assigned to man-mark the Brazilian during the final, but was beaten by him in the air):
"I told myself before the game, 'he's made of skin and bones just like everyone else' — but I was wrong."