The region had slipped to fourth in the pecking order but this victory gives the Ospreys a one-point lead over second-placed Glasgow.
As well as Biggar the Ospreys were indebted to man of the match Tyler Ardron, who scored his side's only try.
Replacement scrum-half Richie Rees scored a late try for Dragons, while Tom Prydie was successful with two penalties but missed a third and as well as a conversion attempt.
Ospreys welcomed back scrum-half Rhys Webb and wing Eli Walker. Webb pulled out of the 22-10 defeat to Scarlets last weekend with illness while Walker had been sidelined with a hamstring problem.
Dragons made two changes from the side which went down to Cardiff Blues with scrum-half Jonathan Evans getting the nod ahead of Rees, while flanker Lewis Evans came in for James Thomas.
Ashley Beck led his side out to mark his 100th Ospreys appearance.
In rainy and windy conditions the opening 25 minutes were a tight affair with no sign of a try being scored by either side.
The scores were locked at 3-3 for most of the half as Prydie and Biggar swapped penalties on five and 14 minutes respectively.
The Dragons did have the edge in the scrum where Dmitri Arhip seemed to have problems.
But it was Arhip and Webb who went close to breaking the try drought on 28 minutes but any score was ruled out by TV match official Gareth Simmonds.
However, an Ospreys try did arrive courtesy of Ardron on 33 minutes. The Canadian latched on to a long pass from Beck after Wales flanker Dan Lydiate had been sent on a run into the Dragons 22 by a finely-timed Biggar pass.
Prydie cut the deficit with a penalty on 37 minutes but, on the stroke of half-time, Ospreys earned a rare scrum penalty which Biggar kicked from inside the 22 to see the home side lead 13-6 at the break.
It took Ospreys only four minutes of the second half to extend their lead to 10 points with a penalty for Biggar who was on target with a fourth penalty on 58 minutes as the home side led 19-6.
The match had something of a twist in the tail when Dragons prop Brok Harris and Ospreys replacement hooker Scott Baldwin were both sin-binned for separate scrum offences.
And a Dragons comeback was threatened when Rees sniped over for a try on 71 minutes.
But Ospreys regrouped with Biggar converting his fifth penalty in the final minute.