Colombia had more than a dozen shots on goal but were thwarted by a combination of poor finishing and stout defending by the Venezuelans, who were without a string of players who tested positive for COVID-19.
The damage was telling for Venezuela in a game where they mustered just one shot at goal compared to Colombia's 17, requiring a dazzling performance from their goalkeeper Wuilker Farinez to keep a clean sheet.
Colombia, who beat Ecuador 1-0 in their opener on Sunday, squandered a string of opportunities and too often mishit final balls into the box against an opponent that defended in numbers.
Miguel Borja thought he had grabbed the winner for Colombia in the 90th minute only to see his effort ruled out for offside and then his side were reduced to 10 men when Luis Diaz was shown a straight red for foul play.
Venezuela have one point from two games but are not out the tournament quite yet, as four of the five teams in each group qualify for the knock-out stages