Montreal was down to 10 players when central defender Adrian (Piscu) Lopez was shown a straight red card at the 44th minute.
After nearly 45 minutes of scoreless play, Heredia was awarded a corner. Cordoba lunged to get a foot on the ball in the six-yard box, poking the ball past Impact goalkeeper Evan Bush, who had given up a short rebound.
"With five minutes left we were going to take a point. It's tough to swallow, especially to give up a goal on a set piece," said Montreal midfielder Davy Arnaud. "The red card changed the game completely."
The teams were tied 0-0 after the first half. Although Montreal had a slight edge on possession — 50.3 per cent to Heredia's 49.7 per cent — the Guatemalan side dominated play with two shots on target.
Head referee Paul Delgadillo pointed to the penalty spot in the 19th minute, but Cordoba's attempt for Heredia hit the crossbar.
Three minutes later, a low blast from 30 yards out by Heredia's David Espinoza skipped off the turf and changed direction on Bush, but he kicked out his feet to make the stop.
More damaging to the Impact was Lopez's red card.
"It's frustrating to lose a game like this when you put a 90-minute effort," said defender Jeb Brovsky. "Playing down a man was not easy but we knew it was not going to be easy with 11 players. And giving up a goal like that it's tough."
Montreal had a great chance to take a 1-0 lead at the 72nd minute when forward Daniele Paploni found midfielder Blake Smith at the top of the penalty box.
A clever flick of the ball over the Heredia defender put Smith in an excellent position to score, but he bounced the ball off the crossbar.
"We were organized in the second half and dealt with the situation we were in pretty well," said Arnaud. "We even got a good scoring chance."
The Impact were missing some key players.
Marco Di Vaio, who scored both goals in Montreal's 2-1 win over D.C. United in Major League Soccer action Saturday, did not travel with the team.
Veteran defender Alessandro Nesta and midfielder Patrice Bernier also stayed behind with the Impact facing a stretch of three games in eight days.
Montreal sits second in Group 5 of the CONCACAF Champions League with three points after a win and a loss. Heredia is first with one win, while the San Jose Earthquakes are last with a loss.
The Impact will play their remaining CONCACAF Champions League matches on Sept. 17 in San Jose, Calif., before hosting CD Heredia on Sept. 24.