Konsel started his professional career with First Vienna and moved to local rivals Rapid Wien in 1985. He then played a part in the successful Rapid team in the 1980s, claiming the League crown twice and most prominently losing the UEFA Cup Winners Cup Final in 1985 against Everton. He stayed at Rapid for 12 years and captained the side in his last two seasons. In 1996 he played in the UEFA Cup Winners Cup Final for a second time, this time against Paris St Germain in Brussels, which Rapid lost again.
In 1997, at 35 years of age, Konsel moved abroad to join AS Roma in Serie A where he was voted goalkeeper of the season in his first year. After an injury-hit sescond year in Rome he finished his career at Venezia.
In 1996 he was chosen Austrian Footballer of the Year. He was voted in Rapid's Team of the Century in 1999, as well as in the AS Roma All Star Team of all-time in 1998.
He made his debut for Austria in an October 1985 friendly match against Yugoslavia and was a participant at the 1990 and 1998 FIFA World Cup. He earned 43 caps, no goals scored. His international farewell match was an August 1998 friendly against France, in which game he was substituted by his eternal rival for Austria's first goalkeeper jersey, Franz Wohlfahrt.