"We needed that and we bowled well and it's nice to back it up with a good partnership with [Franklin] and the boys showed a lot of character today and they're a tough side, obviously. But most important, it's good to have a win and put us back in the tournament."
After winning the toss and batting first, Auckland started off in fine fashion and when Anaru Kitchen (39) and Gareth Hopkins (49) put on 70 for the fourth wicket, the Aces were threatening to make 180.
But, Wellington skipper Grant Elliot took 3-16 with his right-arm medium-pacers and Jeetan Patel (2-24) proved tough to get away for the Auckland batsmen.
Elliot mixed up his bowlers with crafty changes that didn't let the Aces settle in the latter overs and Auckland only managed seven runs from the final 12 balls.
Franklin and Taylor were also busy in the field as Franklin nabbed three catches and Taylor picked up two stumpings with the wicketkeeping gloves.
Auckland may have regretted their decision to only play one recognised spin bowler after left-armer Bruce Martin was nominated to carry the drinks.
Although they lost to Central Districts on Tuesday at the same venue, Wellington's slow men were hard to get away on the dry wicket, which should have been a lesson for Auckland.
Left-arm slow bowler Ronnie Hira was economical for Auckland and took 1-27 from his four overs, but after the visitors spilled a couple of half chances in the field their total looked to be too small as they headed towards their first loss of the HRV Cup.
Michael Papps, who was called in to replace Jesse Ryder after he was ruled out with a calf injury, didn't last long and was bowled in the first over of the Wellington run chase by Auckland seamer Chris Martin.
The loss of Papps brought Taylor to the crease and he combined with Michael Pollard (21) to add 59 for the second wicket and after Pollard played the ball on to his own stumps Franklin and Taylor were brought together in the middle.
The pair added another 48 and when Taylor went in the 14th over the game seemed safe for the Firebirds and Franklin brought the side their first victory of the HRV Cup with a flurry of boundaries.
Auckland, who remain top of the table after winning their first two games, travel to New Plymouth to meet Central Districts on Friday, while Wellington head south to meet Otago in Queenstown on Saturday.