Wellington made steady inroads into the Otago batting order throughout the innings, reducing the Volts to 97-6 at the halfway mark.
That is when Neesham came to the crease, where he formed significant partnerships with Derek de Boorder (31) and Neil Wagner (21). Neesham hit seven fours and a six in his knock of 55, coming at quicker than a run-a-ball.
His knock was eventually ended by McKay and, two overs later, Wellington had Otago bowled out on the penultimate ball of the innings, leaving the Firebirds to chase a challenging 219.
Wellington were given a good start thanks to a half century from Michael Papps but the Otago bowlers took regular wickets, with only one Wellington partnership surpassing 50 runs.
Neesham had already taken three wickets when he was handed the ball for the final over, with the home side needing 12 runs and having only two wickets in hand. Wellington would have fancied their chances with big-hitting former Black Cap James Franklin at the crease, and that feeling only increased after Scott Kuggeleijn sent Neesham's first ball of the over to the fence.
But with four balls left and still seven runs required, Neesham had Franklin caught in the deep for 58, before wrapping up the match with the dismissal of Kuggeleijn the very next ball.
It was Otago's second win from two, sending them to the top of the table and leaving Wellington pointless in last. Each of the other four sides collected two points from the abandonments, meaning Auckland and Canterbury are on Otago's tail, two points back.