In the 7,200-foot altitude of Mexico City, the Crew conceded two early goals, lost defender Eric Brunner to a second caution late in the first half, and contributed two deflected goals off Brunner's central partner Chad Marshall.
Jaime Lozano hit a hat trick -- though one goal caromed into the net off Marshall -- and Marvin Nunez, who assisted on Lozano's opening goal in the fourth minute, scored the second just five minutes later.
After Lozano had tapped in his cross, Dallas product Ramon Nunez blasted in a goal from 30 yards.
Brunner took a second caution in two minutes, leaving his team a man down with seven minutes left in the first half. The Crew, though, raced upfield late in the half, but goalkeeper Yosgart Gutierrez charged out block a shot by Robbie Rogers, released clear by Alejandro Moreno's pass.
Marshall re-directed into his own net early in the second half, when he couldn't deal with Alejandro Vela's cross.
Crew coach Robert Warzycha replaced playmaker Guillermo Barros Schelotto early in the second half., but his replacement Adam Moffat nor subsequent changes could impede a determined opponent.
Cruz Azul kept coming forward, and a seeming stumble by Lozano yielded a phantom penalty that Lozano converted.
Lozano struck again, unexpectedly, when he fired a low shot from the left side that took a sharp deflection off Marshall and dropped past stranded keeper Will Hesmer, who took a pounding through no fault of his own.
Concacaf Champions League
Aug. 26 in Mexico City
Cruz Azul 5, Columbus 0.
Goals: Lozano 4, Nunez 9, own goal (Marshall) 47, Lozano 82, 92+.
Cruz Azul - Gutierrez, Chavez, Brown, Cervantes, Huiqui (Zeballos, 58), Lozano, Torrado, Riveros, Nunez (Lugo, 67) Vela (Hernandez, 63), Villa.
Columbus - Hesmer, Hejduk, Marshall, Brunner, Padula, Carroll, Pirro, Gaven, Rogers (Lenhart, 79), Moreno (Ekpo, 65), Schelotto (Moffatt, 54).
Referee: Oscar Moncada.