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The win pushed the Chiefs level with the Wellington Hurricanes in second place on the table on 45 points. The Hurricanes, who have lost their last three games, play the Auckland Blues later on Saturday.
The Brumbies (30 points) have now dropped out of playoff contention with the New South Wales Waratahs (39) and Melbourne Rebels, who occupy the eighth and final place on 35, ahead of them in the Australian conference.
McKenzie's influence was telling in the tight game, with the utility back sparking, then finishing off a counter-attack to score, while his intercept of Joe Powell's pass set up Johnny Fa'auli's five-pointer in the second.
The Brumbies, who were kept scoreless in the first half, fought back in the second with tries to winger Henry Speight (two) and fullback Tom Banks but were unlucky not to snatch the win.
They had beaten the Hurricanes last week in Canberra and over the last two weeks have not looked the same side that lost six of their seven games in April and May.
With centre Tevita Kuridrani and hooker Folau Fainga'a prominent in getting over the gain line, they had numerous chances to score tries against the Chiefs, but lost the ball in the face of crunching defence.
The Chiefs also showcased how dangerous they will be in the playoffs by being both clinical and unpredictable with hooker Nathan Harris scoring a try following a patient and sustained buildup.
McKenzie then finished off a brilliant counter-attack that had been started following a turnover deep inside their own territory as they went into the break with a 17-0 lead.
The visitors had been trailling 24-5 with about 15 minutes remaining but Banks and Speight both crossed when Jesse Parete was sinbinned for a high tackle to give them a chance of snatching victory.
McKenzie missed a late penalty that would have given the Chiefs an eight-point buffer, but the visitors' mistakes again came back to haunt them and they turned the ball over.