Vettori claimed 3-11 from his four overs, including his three wickets in four balls in the last over.
Two sixes from Grant Elliott's bowling, in the penultimate double value over, carried the Rugby side back into the chase with All Blacks wing/fullback Jordie Barrett smiting two sixes, to score 24 runs to leave eight runs required from the last over.
But it was Vettori who made life difficult by bowling World Cup-winning halfback Andrew Ellis first ball and then former All Blacks wing Marc Ellis was stumped first ball. Former age-group cricket international and Hurricanes Super Rugby player Jason Spice got a run first ball and then Barrett hit out, only to be caught at mid-wicket by Elliott, for 30 from 12 balls.
Rugby needed seven from two balls and while Spice managed a boundary off the last ball it was just short of what was needed for the win.
Team Cricket had batted first and posted 189-5.
It didn't help Team Rugby's Kylum Boshier, who captained New Zealand's Under-19 team at their World Cup in 2018, that his coach, former Test batsman Peter Fulton knew his bowling well.
So well, in fact, that he hit five sixes from consecutive balls bringing up his half-century from 38 balls. Boshier's only over cost him 30 runs.
But there was some satisfaction for the Chiefs loose forward when he took the catch that ended Fulton's innings on 80, scored off 51 balls.
That was largely down to a fine bowling spell by Andy Ellis, who took three wickets for 15 runs from three overs.
As part of the game one over was set aside to provide double value for the runs scored and Team Cricket managed 22 runs as Grant Elliott took advantage of Mat Sinclair's bowling. Sinclair, the former Test cricketer, having been called into the Rugby side as a late replacement.
Two-time World Cup-winning captain and flanker Richie McCaw opened the bowling and utilised some swing from his left-arm medium pace to concede only 10 runs from his two overs but bowling the penultimate over, he had another 10 runs taken to end with 0-20.
Elliott ended with 34 not out, scored from 13 balls.
Team Rugby made a strong start with Will Jordan and Ofisa Tonu'u putting 75 runs together before Tonu'u, who unleashed several reverse sweeps in scoring 29 from 28 balls was dismissed.
But the power in the start came from the recent All Black wing/fullback Will Jordan who powered his way to 49 from 29 balls, especially taking toll of former New Zealand allrounder Jacob Oram's bowling, before bottom-edging a wide ball from Craig McMillan which was taken by wicketkeeper Adam Parore standing up.
In the same over Sinclair went in similar fashion with McMillan taking 2-25.
Boshier and Jock McKenzie came together and profited from Jordan Watson's charitable bowling to bolster the scoring rate. They added 48 runs but departed on successive balls from former New Zealand off-spinner Nathan McCullum, who finished with 3-20. Boshier was yet another victim to Parore catching standing up after scoring 31 off 13 balls.
With the run rate mounting, Elliott started his first over by conceding a six over square leg to Brad Weber but then holed out to mid-on where McCullum held the chance.