Earlier, Wellington only set a modest total as Auckland's bowlers did well to contain the home side to 157-6.
Maybe it was down to fatigue having played the night before, but Wellington struggled to get any momentum to their innings as they fumbled their way through their 20 overs.
No 3 Michael Papps top-scored with 59, while Stephen Murdoch and Woodcock each chipped in with scores in the 30s.
Wellington lost wickets with the final three balls of the innings as former Black Cap Michael Bates found himself on a hat-trick, then Jeetan Patel was run out with the last ball of the 20th over.
Despite hitting a six off the last ball to guide Wellington to a win over Central Districts last night, young batsman Henry Walsh was only at the crease for a couple of minutes at the end of the innings but didn't face a ball.
Walsh will no doubt get another chance as the competition progresses.
Auckland 158-4 (Luke Wright 56, Craig Cachopa 42*; Brent Arnel 3-25)
Wellington 157-7 (Michael Papps 59, Stephen Murdoch 35; Luke Wright 2-25, Michael Bates 2-33) by six wickets.