Striker Tressor Moreno scored the winner with a twice-taken penalty in the 22nd minute after Venezuelan goalkeeper Gilberto Angelucci was harshly adjudged to have brought him down as he burst into the box.
Moreno, Colombia's best player in the first half, had to retake the kick after Brazilian referee Marcio Rezende de Rezende disallowed his first effort because his team mates had encroached into the penalty area.
Venezuela have only one Copa America victory to their name against Bolivia in 1967, but were hopeful of making a good start in Group A of the 12-team tournament after an unprecedented run of good results in World Cup qualifiers.
They dominated the second half and created plenty of chances but were repeatedly denied by Colombia goalkeeper Juan
The "Red Wines" almost snatched an equaliser from a swerving Juan Arango free kick in the 85th minute, which was turned around the post by Henao with a low dive to his left.
The Venezuelans also had a goal disallowed for offside early in the second half and Colombia were lucky to survive a lacklustre display that did not match up to their slick performance in the first 45 minutes.
Their only real chance after the break came in the 88th minute when David Ferreira passed the ball across the open goalmouth from the left flank to an unmarked Jairo Patino, who fluffed an easy opportunity from close range.
Ferreira also produced one of the moments of the match in the 45th minute after a brilliant run down the right wing, only to fire the ball straight at Angelucci from five metres.
Colombia play Bolivia their next match on Friday and hosts Peru meet Venezuela.