However it was the Champions who had made the more confidently start Shaun Connor opening the scoring with a penalty after only three minutes.
However the Osprreys soon were back in the contest as Charlie Hore, filling the number 10 role in place of the sidelined Gregor Townsend, levelled the scores with a penalty.
With the Reivers now enjoying the bulk of possession and territory, Hore collected another three points after an award for offside. Play became bogged down in midfield for a lengthy spell and it was the home troops who tried to raise the tempo in the build up to the break.
They carved out several promising openings, but only had another Hore penalty to show for their efforts – until the final move of the half. Hore was blocked a couple of paces short and Test winger Nikki Walker was ideally positioned on his shoulder to finish clinically. Hore followed up with the conversion to widen the gap to 13 points.
The rattled Ospreys showed much more urgency after the restart, but could not make immediate inroads through the well-marshalled home defence.
Borders fans roared their approval when Semo Sititi came off the bench in place of Andy Rennick. The Samoan international skipper had been forced to delay his return because of red tape associated with his journey to the UK.
The Welshmen made a spate of substitutions in a bid to change the pattern of the game and they gave themselves a glimmer of hope when Connor clipped over his second penalty from 40 metres following a line-out infringement.
But it was the Reivers who appeared to be the stronger outfit when it mattered in the final 10 minutes.