New arrival Dominic Oduro scored the only goal of the game for the Impact inside five minutes but Frank Klopas watched his side concede two penalties and have a man sent off before the final whistle.
Cruz Azul were awarded the first of their two penalties after half an hour but Montreal goalkeeper Evan Bush saved the spot kick to keep the Canadian-based MLS side in the lead at halftime.
The drama increased in the second-half as Justin Mapp hit the post for the Impact before Jack McInerney and a Cruz Azul player were both sent off on 75 minutes.
Five minutes after the double sending off, the Mexican side were awarded their second penalty of the game but missed again as the post came to Montreal’s rescue.
The Impact held on to record their second win of preseason and now prepare to face another Liga MX side in Pachuca in their CONCACAF Champions League quarterfinal next Tuesday.
The first-leg of that tie will be played in Mexico before the return leg in Montreal the following Tuesday, just three days before the 2015 MLS season is set to kick off.