They collected only 29 off their final five overs as Auckland worked their way back into the game.
Adams played a big hand in that, finishing with four for 25 from his four overs.
Auckland's chase got off to the worst possible start when Lou Vincent was run out he tried to sneak a single off the first ball of the innings.
English import Phil Mustard and Gareth Hopkins then struggled to build momentum against tight bowling from Mark Houghton, Dimitri Mascarenhas and Dane Hutchinson.
The required run rate climbed steeply - they needed 127 off the last 10 overs - until Mustard, who had looked completely out of sorts, sparked into action with some powerful hitting and was left stranded on 97 not out off 66 balls.
He found good support from Craig Cachopa (24 off 19), but the target always seemed beyond them.
The result put Wellington with Northern Districts on top of the table, although ND have a game in hand.