Skiverton moved up from the back for a left-wing corner by Arron Davies and got down low at the near post to head the only goal of the match.
Oldham headed to Somerset on the back of a good run, but their lethal attack duo of Chris Porter and Paul Warne were well held by a defence in which Skiverton again excelled and Terrell Forbes - who was with Oldham last season - proved equally effective alongside the skipper.
The match was always tight and tense and for a long spell in the first half goals were seldom threatened. It was 32 minutes before the Latics forced a save from home goalkeeper Steve Mildenhall, another who had a spell with Oldham in the past.
Kelvin Lomax, on as an early substitute for the injured Gary McDonald, got in a good cross from the right. Warne did well to reach it, but his flick was straight at the keeper.
Yeovil, better at defending than scoring these days, began to show more as an attacking force and twice within a minute late in the half their powerful young midfielder Chris Cohen got to the edge of the box' but was over the bar with his drives.
Skiverton's goal set the scene for an equally entertaining second half, with Latics pushing forward but only on rare occasions troubling Mildenhall.
Seven minutes into the half Yeovil brought on Wayne Gray in place of Lee Morris, who was struggling with an injury, and Gray proved their dangerman.
In the 71st minute he flung himself at a Davies free-kick but his header went wide of the post. And in stoppage time he shrugged off two defenders in a powerful run that took him into the box, but his final shot did not match the approach and Les Pogliacomi saved well at the foot of his post.
The result left Yeovil second and in their highest ever place in the football pyramid.