Salomon Rondon, Venezuela's biggest threat in the match, headed the winner past goalkeeper David Ospina and into the far corner in the 60th minute.
It was Venezuela's first victory over Colombia in the Copa America, a major step for a country known more for producing baseball players than footballers.
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'We knew we were facing one of the favourites, and that we had to be disciplined and not let them think,' Venezuela coach Noel Sanvicente said. 'This game should put a smile on every Venezuelan.'
Colombia, a quarter-finalist in last year's World Cup and one of the tournament's most exciting teams, was poor throughout the Group C match, despite fielding the likes of Real Madrid's James Rodriguez and Radamel Falcao of Monaco.
'We couldn't play the way we usually play,' Colombia coach Jose Pekerman said. 'We had difficulty finding the correct passes and Venezuela defended well.'
Venezuela frustrated Colombia's attacks before they could get going, put up a strong and physical defense, as well as always threatening on the counter-attack.
Venezuela's Ronald Vargas had the best early chance, with a point-blank shot in the 27th minute that was stopped by Ospina. Alejandro Guerra also came close in the 42nd minute with an effort that sailed just wide.
By contrast, Colombia offered little in the first half.
Pekerman's team attacked early in the second half and, just as it was building momentum, Rondon broke the deadlock with the goal - 10 minutes after he had hit the crossbar with another header.
Venezuela keeper Alain Baroja came up with his two key saves of the match in the 79th minute, when he denied James and Edwin Cardona as Colombia pressed for an equaliser.
Group C action continues when Brazil plays Colombia on Wednesday and Peru faces Venezuela on Thursday.
Rodriguez (right) tries his luck at goal as Colombia search desperately for an equaliser on Sunday
Read more: http://www.dailymail.co.uk/sport/football/article-3123829/Colombia-0-1-Venezuela-Solomon-Rondon-s-header-provides-early-Copa-America-shock.html#ixzz3d5ywGe1j
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