The Albicelestes remain on top of the South American qualifying zone but saw their lead cut top three points by Colombia.
Sergio Aguero opened the scoring with a penalty in the third minute before Segundo Castillo levelled the match after 16, with Argentina midfielder Javier Mascherano sent off in the closing stages.
Argentina became the only side not to lose in Ecuador so far in the qualifying campaign, Messi playing the last 30 minutes as a replacement for Aguero.
Argentina have lost only once in the current campaign - 1-0 in Venezuela in their second match - but have drawn all of their last three games, something which has frustrated Messi.
The Barcelona forward told Channel 7: "We had what it takes to win the game, but we are getting used to draws.
"We failed to take profit from free space in the first half, but we are close to the qualification. That is what matters.
"We scored the first goal very quickly. To play here is very tough. Then they equalised."
Manchester City forward Aguero said he was not as tired as he had expected after playing at 2,800metres above sea level.
He said: "I found it hard to breathe very quickly, but finished fine. I thought playing at this altitude would be more difficult.
"The Ecuadorians are physically strong and are very tough to beat here. We suffered quite a lot."