But defeat for Edinburgh, who had Nick De Luca sent off late on for a tip tackle on Tom Grabham, means they remain winless in the league since a trip to Connacht on December 1.
The Ospreys made the perfect start to the evening when they surged forward and Allen crossed the whitewash following a cute offload from Jonathan Thomas. Fly-half Matthew Morgan added the extras and the home side found themselves 7-0 up after five minutes.
Edinburgh had the chance to reduce the deficit on 12 minutes when the Ospreys were penalised for not releasing after a break by centre Ben Atiga, but fly-half Harry Leonard was wayward off the tee.
And the Ospreys kept their try line intact despite several sniping attacks from close range, led by De Luca and Leonard as Edinburgh tried to build some pressure.
Ospreys gradually worked their way upfield and their second try on 21 minutes was a repeat of the first, Allen touching down from close range after a penalty was kicked to touch. Morgan made no mistake with the conversion to give his side a 14-point lead.
Ospreys threw wave after wave of attacks at the Edinburgh defence in the last ten minutes of the first half, with the visitors' tackle count mounting ever higher.
But on the stroke of half-time 21-year-old debutant Hamish Watson brought Edinburgh back into the game when he finished off a flowing backs move on the left after a great break down the other flank from Netani Talei and Dougie Fife.
Leonard added the extras to half the deficit, but Morgan was given an easy chance to strike back immediately with a penalty in front of the Edinburgh posts only to snatch at the chance and send it wide.
Edinburgh had the first shot at goal after the break on 43 minutes when Cai Griffiths was pinged for not rolling away from the tackle, but Leonard missed the left hand upright by inches.
After a frantic ten minutes during which possession shifted from side to side Morgan was given the chance to add to the Ospreys lead after Edinburgh were again penalised for hands in the ruck. But the 20-year-old half-back's effort faded away across the posts and the gap remained at seven points.
But he made it third time lucky from close range after Edinburgh were pinged at the scrum to give the Ospreys some breathing space. The visitors' job was made harder by the sin-binning of Atiga, who tried to play the ball on the floor.
And the one-man advantage paid immediate dividends after Richard Fussell led a quick break down the left, the ball was recycled quickly and scrum-half Rhys Webb dotted down from the base of the ruck. Morgan added the extras and suddenly the Ospreys looked comfortable at 24-7.
The error count mounted as the Ospreys went in search of a bonus point in the latter stages, opting to keep the ball alive rather than kick for the posts.
With four minutes remaining De Luca upended Grabham near the left touchline giving referee David Wilkinson no option but to reach for the red card.
Ashley Beck was stopped two metres short of the Edinburgh line but his offload out of the back of the hand went to an Edinburgh player late on as the bonus point proved elusive for the Ospreys.
|Ospreys Score Card|
|Edinburgh Rugby Score Card|