Deportivo started the match the stronger and scored the opener in the 37th minute through Argentinian forward Jeremias Caggiano on Tuesday.
Gabriel Penalba netted an ill-deserved equaliser for Tigre in the 66th minute and 10 minutes later, youngster Leandro Leguizamon came off the bench to head the winner from a free-kick.
Leguizamon told Fox Sports: "My goal is important to me but more to my team. It was a hard match until that moment.
"Anzoategui run a lot and high balls were one of their strong points. We'll try to win the second leg too."
Disappointed Anzoategui forward Rolando Escobar said: "The team tried to get the job done but the equaliser shocked us and Tigre stepped forward and scored again."
The two teams are battling to qualify for Group Two, where the winners will play Brazilian giants Palmeiras, Peruvian champions Sporting Cristal and Paraguay's Libertad.
In Mexico, Chilean side Deportes Iquique were on course for a great win in their Copa Libertadores debut until hosts Leon made it 1-1 with only two minutes remaining.
Argentinian midfielder Rodrigo Diaz opened the scoring in the third minute with a potentially vital away goal for Iquique but, with Cristian Diaz's side celebrating the win, Nelson Sebastian Maz came off the bench to head in a late leveller.
Maz had the opportunity to net the winner in injury time, but Iquique keeper Rodrigo Naranjo saved his shot.
Uruguayan giants Penarol, Ecuador's Emelec and Argentinian champions Velez Sarsfield await one of these two teams in Group Four.
The second legs will be played next Tuesday.