Canobbio started his career in 1988 by joining CA River Plate. By that time the team was in a very difficult situation concerning both sporting and economic matters. Then, after some bad performances, River lost its category and was relegated to Segunda División Uruguay in 1990.
In 1991 River achieved the title and was promoted to Primera Division with a significant contribution from the young striker. Osvaldo also took part in River's all-time best squad in 1992 when his team reached the 2nd position in the Uruguayan First Division tournament (Primera División Uruguaya).
By 1994 Canobbio had joined Nacional, achieving good performance. He then moved to Argentina in 1996 and started playing for Deportivo Español. In Argentina he also joined Talleres de Cordoba (in 1997) and Racing Club (in 1998) where he became the highest goalscorer of the team.
In 2002, after some years playing in Argentina, Osvaldo moved to Chile and joined Cobreloa and participated in Copa Libertadores. According to the sport press in Chile, Canobbio was considered one of the strongest reinforcements from abroad.
After almost fourteen years playing in South America, Osvaldo moved to China to join Yunnan Hongta, a defunct team which used to participate in the Chinese League.
Since 2004 until 2007, Osvaldo has played for his former team, River, in the Uruguayan First Division. Then he played for C.D. Olimpia in the Liga Nacional de Honduras. He was recently hired by Liverpool F.C.