A Sam Wilson penalty had visitors City in front at half-time but substitute Osborne levelled matters midway through the second period with a ripper of a strike from 25 yards.
Teed up by Micky Hubbard, Osborne unleashed an unstoppable effort across the keeper and into the top corner of the net.
Osborne’s drive was the Hammers’ first goal of the Wessex Premier campaign and allowed them to avoid a third consecutive defeat following losses to Horndean and Brockenhurst.
Boss Kemp was satisfied with the end result and praised his players for their physical exertions during the course of the game.
He told the Daily Echo: “I thought it was a fair result over the 90 minutes and it was a vast improvement from the way we played in the first two games.
“It was important for us to play well and get a point on the board after a couple of defeats.
“Winchester had a couple of great opportunities to win the game in the second half but we also had two or three half- chances.
“It was a quick tempo and a really good advert for Wessex League football. At the end of it, our boys had given everything.”
Sitting 18th in the standings, Kemp’s men are next in action tomorrow when they travel to Bemerton Heath Harlequins (7.45pm).
Bemerton, who beat Christchurch on Saturday, are currently ninth in the table having picked up four points from their first three games.
Hammers: Ackerman, Allen (Osborne, 55), Blackburn, Cole, Cook (Price, 70), Donachy, Hubbard, Legg, Penny, Warren (Martin, 65), Yelland.