Edinburgh Gunners ended Connacht's slim hopes of making the Heineken Cup qualification play-off as they ran in three tries at the Sportsground.
Pens: McHugh 3
Tries Dickinson, penalty, Di Rollo
Cons: Paterson 2
Connacht led 10-3 at half-time through a Tom Tierney try and a Mark McHugh conversion and penalty.
But Edinburgh hit back when Alasdair Dickinson went over and added a penalty try after good forward pressure.
The Gunners added the decisive score in the 71st minute when Marcus Di Rollo went over and Chris Paterson converted.
It was Connacht's first home defeat in six games stretching back to December.
Connacht: D Slemen, M Mostyn, J Hearty, G Williams, D Yapp, M McHugh, T Tierney, D McFarland, J Fogarty, S Knoop, D Gannon, A Farley (capt), J Muldoon, M Lacey, C Rigney.
Replacements: J Merrigan, A Clarke, D McGowan, M Swift, C O'Loughlin, P Warwick, C McPhillips.
Edinburgh: H Southwell, C Paterson (capt), M Di Rollo, P Jorgensen, F Leonelli, P Godman, R Lawson, A Jacobsen, A Kelly, A Dickinson, A Kellock, D Duley, M Mustchin, S Taylor, A Strokosch.
Replacements: K Traynor, D Hewett, F Pringle, A MacDonald, M Blair, A Monro, M Pyke.