Full-back Carl Bezuidenhout kicked 14 points and while Jeff Hassler's second try of the match on 61 minutes brought the Ospreys to within three points, Edinburgh closed out the win.
Dan Biggar did secure a losing bonus point for the Ospreys with his fifth penalty of the night a minute from time but Edinburgh now find themselves within touching distance of the top six in the RaboDirect PRO12 in Solomons' first season in charge.
"The crowd definitely helped us. The players could feel it and produced an outstanding performance against a strong Ospreys team," he said.
"We continue to improve and this team is really coming together.
"I'm very pleased with the win. It was a good performance against a strong Ospreys side and was a real team effort."
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