Kavanagh led 13-10 at the break and looked safe when it scored its third try with 10 minutes left to lead 23-17.
But Menzies, in its first year in the Highlanders First XV competition, came storming back and scored a try with four minutes left.
It was desperate defence by Kavanagh College in the final minutes. With only seconds left a Menzies College player was about to fall over the line for the winning try, when the ball was knocked out of his hands in a desperate tackle.
The key to the Kavanagh win was the rugged defence of the loose trio of James Tomkinson, Tom Bleach and Nathan O'Brien.
Lock Mason Grubb dominated the lineout and Guy Woodhouse had pace in the midfield.
The accurate goal-kicking of fullback Rhys Johnson counted in the end when he kicked two penalty goals and a conversion.
Halfback Tom Docherty was a scoring machine for Dunstan when it outplayed John McGlashan 44-12 at Alexandra.
Docherty scored two of Dunstan's six tries and kicked five goals to score 22 points, and he teamed effectively with Keelan Baird and Sam Galletly.
Loose forwards Paddy Rabbitt and Kyle Harris dominated the ruck and maul.
South Otago led 12-7 at halftime and went on to score five tries and beat St Kevin's 29-14 at Oamaru.
Props Caleb Edwards and Duncan Leslie gave South Otago a dominant scrum and loose forwards Johnny Sergeant and Elliott McKinnell dominated the breakdown.
Gore suffered its first loss of the season, when it was beaten 30-10 by Otago Boys' at Littlebourne in Dunedin.
Otago Boys' was still feeling the effects of the hard interschool game against Timaru Boys' midweek and looked tired in the first spell when it held a slender 8-7 lead.
Gore's forwards were strong with the pick and go in the first spell.
Otago Boys' came to life in the second spell and scored four more tries to win comfortably.
Southland Boys' leads the competition after seven rounds with 34 points and is followed by Gore 31, Waitaki Boys' 30, Otago Boys' 29, King's 29, Dunstan 23, Menzies College 17, Kavanagh 16, South Otago 11, John McGlashan 7, St Kevin's 5, Verdon 0.
Otago Boys' has two games in hand and remains unbeaten.
Waitaki Boys' beat Verdon College 51-0 and Southland Boys' beat King's 18-8 in a midweek game.