But American wing Dirksen then stole the show midway through the second half, racing on to his own chip before outpacing the Edinburgh defence to score what turned out to be the winning try as Dan Biggar kicked the vital conversion.
It was the visitors who got off to a stronger start and home fans had their hearts in their mouths when a box kick was charged down but fortunately for the Murrayfield faithful it ammounted to nothing.
But after this early pressure the Gunners settled and Godman had the first chance to get some points on the board with a penalty, but the fly-half sent his kick horribly wide on 11 minutes.
However he made no mistake three minutes later as he got another chance, this time right in front of the posts, making no mistake.
And on 19 minutes the hosts opened up an eight-point lead as centre Fife went over for a debut try after some good work from Chris Paterson and Netani Talei.
Godman however sent his kick wide of the posts and he was soon made to pay for his wayward kicking as the Ospreys drew level.
On 22 minutes Biggar kicked over a drop goal and just four minutes later it was 8-8 as hooker Hibbard bundled his way over, with a little help from his fellow forwards.
Biggar followed Godman's lead and was wide with his conversion and this gave Godman to chance to send his Edinburgh side in at half-time ahead when the Ospreys were penalised right before half-time, the fly-half making no mistake for an 11-8 lead.
The home fans were on their feet ten minutes into the second-half as they welcomed new signing Ulises Gamboa onto the field as he replaced prop Jack Gilding.
And the Gunners players on the pitch were clearly buoyed by the Argentinean's presence as Godman kicked a 54th minute penalty to extend the Edinburgh lead.
However, just two minutes later Dirksen brightened up the game with a stunning solo try, Biggar with the easy conversion to hand Ospreys the lead for the first time in the game.
On the hour Godman had a simple penalty kick for Edinburgh to retake the lead but he somehow missed from right in front of the posts.
But luckily for him Biggar was wide with a penalty of his own two minutes later to keep Edinburgh within penalty distance of a win.
However, despite some good possession it just wasn't to be for Edinburgh and the Ospreys collected the narrow victory.
|Edinburgh Rugby Score Card|
|Ospreys Score Card|