King showed stellar form with the ball, taking two wickets and conceding just 12 runs in 10 overs as the visitors were held to 154. Rob Hatchard (4-28) provided more ammunition.
After a rocky start, Wimborne’s Tom Caines (43) grafted his way through before Ben Ratcliffe (23) joined James Crease to put on 41 for the fifth wicket.
Quick-fire wickets soon stifled the momentum, however, before James Williams (30*) at least helped the Minstermen to their respectable total.
In response, openers Matt Thomas (42) and Mike Salmon (32) put 68 on for the first wicket as Swans took control and it looked like it would be a comfortable victory as they reached 115 for one.
The tide quickly turned, though, when spin-bowler Harry Beardsley (6-26) was handed the ball and swiftly reduced the hosts to 118 for five.
But up stepped King (14*) with a calm, assured tail-end performance and helped his side over the line in a nail-biting climax which saw Swanage win with only one ball to spare.
“It was always going to be a low-scoring game on a wicket that was wet and green,” commented Wimborne captain James Miller. “I was happy with 154 as I felt with our bowling attack we could restrict Swanage.
“Matt Thomas batted very well for them and ended up playing a match-winning innings, but the star of the show for me was Harry Beardsley, who stepped up to the plate after begging me to be picked for the first team all week.
“We were struggling and were facing a heavy defeat until Beardsley bowled a master class of very slow off spin.
“With figures he got, I think he has secured his place in the team for at least the next couple of weeks.”
It was Wimborne’s first defeat of a rain-affected campaign while Swanage moved up to third in the early standings.