With the scores level at 23-23 after 80 minutes, and after McKenzie had missed a dropped goal and an earlier penalty, the All Blacks fullback nailed a 45m effort to seal the dramatic win.
The victory reverses the result from the last time the two teams clashed, a second round match in February won by a dramatic second-half comeback from the Highlanders.
The Chiefs led 10-5 at halftime after a scrappy first 40 minutes that saw a slew of penalties and opportunities missed.
Etene Nanai-Seturo crossed for an early try by the visitors, acrobatically diving into the corner having set up by halfback Brad Weber.
The Highlanders hit back almost immediately through flanker James Lentjes after a clever bit of play as he dummied to pass out of the back of a maul, tricking five Chiefs defenders into thinking he was going to the right before scooting round the left to dot down.
They drew level early in the second half with a second try from co-captain Aaron Smith as he benefitted from loose forward Shannon Frizell's excellent line break.
The Chiefs continued to look the more dangerous in attack and after landing two penalties following incessant pressure, they grabbed a second try after a fine angled run from prop Angus Ta'avao saw him crash over as they extended the lead to 10 points.
The Highlanders found some momentum in the closing stages though and an excellent maul towards the Chiefs line eventually saw the backline released and replacement first five-eighth Josh Ioane crossed to score next to the posts.
He then levelled the score with a penalty with three minutes remaining, before McKenzie missed a drop-goal attempt and a penalty from the halfway line.