The home side posted 224 for nine, which seemed like it would not be enough.
Wellington threatened to quickly overrun the total with Sophie Devine blasting a quickfire 45 at the top of the order.
But the Sparks were able to claw back the run rate through the middle period of the innings while chipping out wickets.
Beth Langston, Suzie Bates and Sophie Oldershaw grabbed two wickets apiece as the visitors slipped from 73 for none to 167 for six.
White Ferns spinner Leigh Kasperek came back on and took two late wickets to help seal the victory.
But it was Bates who struck the coup de grace, bowling No 11 Wheturangi Charteris midway through the 48th over.
"It was tense game wasn’t it?" a relieved Sparks coach Nathan King said.
Otago had lost its previous three one-day games and its two twenty20 matches in a disappointing start to the campaign.
"The reality is we have played really poorly in the first five games. [Yesterday] was an improvement and it is amazing what a win can do for the team morale and confidence."
Otago produced a much improved batting effort to set up the win.
Kasperek top-scored with 46 from 53. Polly Inglis made 33 at No 3.
Beth Langston helped hold the middle order together with a valuable innings of 37. And Katey Martin showed some tenacity during her knock of 37.
The innings had got off to a poor start, though, with Bates scooping a catch to short midwicket in the second over.
Wellington, by contrast, made a blistering beginning. Devine seemed particularly keen to hit the ball as hard as she could.
She swatted a six and consecutive fours from Georgia Clarke’s third over. But when she was bowled by Bates, her side started to lose its way.
Wellington recovered from 41 for four to win Saturday’s one-dayer.
Amelia Kerr steadied the innings with a fine knock of 74. Eimear Richardson (55) and Jess Kerr (42 not out) put on a damaging stand of 68 for the ninth wicket to help their side post 260 for nine.
Bates took three for 36 but missed out with the bat. Inglis (34) and Kasperek (28) got the chase back on track briefly, and Megan Gibbs added 28.
But the Sparks were dismissed for 169 with Amelia Kerr and Lucy Doolan claiming three wickets apiece to help secure the 91-run win.
• In other games, Auckland beat Canterbury twice and remains unbeaten.
Central Districts and Northern Districts shared the honours with one win each.
