It took Henson just three minutes to have an impact on the match, putting over a sweetly struck penalty kick from 40 metres out.
But though the Ospreys dominated possession and territory they made heavy weather of getting points on the board.
In his first game for seven weeks since being suspended for an elbow on Leicester prop Alex Morenjo, Henson was looking to put a big performance in front of stand-in Wales coach Scott Johnson.
Henson was out to impress nine days before Wales travel to Ireland for the third round of RBS 6 Nations Championship.
The 24-year-old, who started at full-back, looked sharp but the odd mistake crept into his game, most notably putting a penalty kick to touch past the dead ball line.
After Borders outside-half Charlie Hore put the Scottish region 6-3 in the lead with penalties on 13 and 24 minutes, Henson levelled matters with a penalty kick from inside the 22.
With the amount of possession the Ospreys enjoyed they should have opened up a comfortable lead.
The home side looked as if they had a try on the 30-minute mark but flanker Richie Pugh was held up over the line.
Following a break by Henson down the right flank, the ball was recycled for Shaun Connor to put over a simple drop goal from in front of the posts to give the Ospreys a 9-6 half-time lead.
The Ospreys upped their efforts after the break, scoring 10 points in a two-minute spell.
First Henson kicked his third penalty six minutes into the half.
That was followed by a fine team try, started by some inter-passing between the forwards and ending up with Henson putting centre Andrew Bishop over on the left.
Henson converted to give the Ospreys a 19-6 advantage.
The Ospreys went back into their shells again after that, but had Pugh clung on to an interception they could well have celebrated a second try.
With an injury to wing Steffan Terblanche and Adrian Cashmore going to full-back it allowed Henson some game time at number 12 – the position he hopes to figure in for Wales.
But though Henson converted a fourth penalty four minutes from the end it was the Borders who finished the stronger.
With the Ospreys having two players sin-binned – Cai Griffiths and Alun Wyn Jones – it opened up for the Borders, who scored two late tries through replacement scrum-half Johnny Weston and full-back Calum MacRae.
That made Henson’s kicking contribution all the more important to the Ospreys’ four-point win.