De Luca got his marching orders from Irish referee David Wilkinson for a tip tackle on replacement wing Tom Grabham, but even with the man advantage the Ospreys couldn't claim what would have been their second try bonus point of the season.
The win enabled them to overtake the Scarlets and return to the top four on a night when Glasgow Warriors beat leaders Ulster to narrow the gap at the top to three points.
Ospreys former Wales Under 20 No 8 Morgan Allen powered for two first half tries, both converted by Matthew Morgan, to give the champions a 14 point lead in the opening 22 minutes. Hamish Watson pounced on the stroke of half-time for the visitor's only try and Harry Leonard's touchline conversion cut the gap to one score at the break.
Morgan missed with two simple kicks either side of the interval before finally increasing the lead with a penalty after 54 minutes. The Ospreys were spluttering in their pursuit of the bonus point, b t when centre Ben Atiga received a yellow card for a ruck offence the home side made the most of their numerical advantage.
Within a minute they had worked scrum half Rhys Webb over for a try which Morgan improved. Edinburgh's defence did enough to stop them from adding a fourth, for which Scottish neighbours Glasgow Warriors might be eternally grateful as they chase a home tie in the play-offs.
Scorers: Ospreys: Tries: M Allen 2, R Webb; Cons: M Morgan 3; Pen: M Morgan
Edinburgh: Try: H Watson; Con: H Leonard