The Ospreys' woes continued as Edinburgh marked new Scotland coach Frank Hadden's last game in charge with a first league win over the champions.
Edinburgh (10) 24
Tries: Dickinson, Taylor, Dewey
Cons: Paterson (3)
Ospreys (10) 18
Tries: Terblanche, M Jones
Con: M Jones
Pens: M Jones (2)
The visitors enjoyed a dream start with Stefan Terblanche's second-minute try, but Edinburgh's pack took control.
Alasdair Dickinson and fit-again Lion Simon Taylor crossed, before Matthew Jones' try kept the Welsh in the game.
Rob Dewey crashed through weak Ospreys tackles for the decisive score, Jones' late penalty saving a bonus point.
Murrayfield looked inviting to the Ospreys at the start, with the outstanding Sonny Parker breaking from deep in his own half.
He linked with Andrew Bishop who sent Terblanche clear in the corner.
Jones failed with two penalty attempts, but Edinburgh gradually got into the game as their pack outgunned the visitors and mounted a sustained campaign in the Ospreys' half.
A kickable penalty was sent into touch, and from the line out tight-head Dickinson battered over.
Chris Paterson and Jones exchanged penalties to leave the scores level at the break, and it was the Gunners who came out firing in the second period.
Number eight Taylor marked his return from injury with a short-range try, popping up on the shoulder of Mike Blair to breach a ragged Ospreys defensive line.
The visitors used Jonny Vaughton's pace to hit back, the wing hauled down by Hugo Southwell yards short of the line but placing the ball for the supporting Jones to pick up and score.
Edinburgh's powerful centre Dewey then blasted through the Ospreys line for a converted try to take the Gunners' lead to 24-15.
That should have been increased when Blair went on a clear 75m run to the line after turnover ball from an Ospreys attack.
But Shaun Connor hauled him down on the try line and the scrum-half knocked on to blow a certain score.
There was no way back for the champions, though, who needed Jones' boot to land the last kick of the game to finish within seven for a bonus point.
Ospreys coach Lyn Jones:
"We're missing nine players and teams are targeting us as the league champions.
"It'll be a difficult couple of months, but we're building towards December, and this game was a big step forward from the Borders loss.
"We played too loosely and kicked poorly against a side who are close to being a Scotland international outfit, but the youngsters are learning and at least we got a bonus point."
Edinburgh: Southwell, Paterson, Di Rollo, Dewey, Webster, Godman, Blair, Jacobsen, Hall, Dickinson, Kellock, Murray, Mustchin, Taylor, Hogg.
Replacements: Hodge for Godman (73), Lawson for Blair (64), Kelly for Hall (67), Smith for Dickinson (69), Pringle for Kellock (57), Cross for Mustchin (63). Not Used: Leonelli Morey.
Ospreys: Terblanche, Mustoe, Parker, A. Bishop, Vaughton, M. Jones, Spice, D. Jones, Bennett, Millward, Cockbain, Newman, Bateman, Thomas, Tandy.
Replacements: Connor for Terblanche (25), B. Williams for Bennett (66), James for Millward (75), Evans for Bateman (73), Beech for Thomas (69). Not Used: Wells, Brew.
Referee: Alain Rolland (IRFU)