But McKenzie denied his side had been worked out and maintained execution errors at key moments were more to blame against the competition's most frugal defensive side.
"I reckon if [we'd] played like that last week we would have won."
Cooper, among the Reds' best, answered several of Wallabies coach Robbie Deans' criticisms in a lively attacking display. He also kicked well.
In Durban, Morne Steyn kicked all the Bulls' points, including a victory-clinching penalty five minutes from the end, to beat the Sharks 18-16.
Steyn's six penalties makes him the competition's leading scorer as the Sharks were denied after halfback Charl McLeod's try and a touchline conversion from Pat Lambie gave them the lead in the last 10 minutes.
The win takes the Pretoria team back to second overall and five points clear of the Cheetahs in the South African conference.
-AAP