Edinburgh failed to make it five successive victories as an Irish rolling maul once more haunted a Scottish side.
Edinburgh (0) 13
Pens: Godman, Laney
Connacht (10) 16
Pen: Elwood 3
The home side had dragged themselves back to equality and looked poised for victory when Connacht's maul won a penalty, kicked by veteran Eric Elwood.
Edinburgh had the early pressure but found themselves behind to a try from Dave Slemen on the counter attack.
Craig Smith replied after the break, but Connacht had the final say.
Edinburgh, with Simon Webster the only member of last week's successful Scottish side to be involved, pressed early on.
But the winger's knock-on on at the halfway line led to visiting scrum-half Chris Keane dashing to the Edinburgh 22, where a penalty was awarded and kicked by Elwood.
A turnover deep in Irish territory was slipped to the left wing and the move ended at the other end with number eight John O'Sullivan kicking ahead and left winger Dave Slemen being first to the ball for the try, converted by Elwood.
Missed penalties meant Edinburgh went into the break 10-0 behind.
And their over-eagerness continued in the second half when Phil Godman's pass was intercepted, with Ally Kellock subsequently red-carded and Elwood kicking the penalty.
Edinburgh pulled back seven points when Brendan Laney entered the line to start a long phase of passing and then set up Smith to score.
Godman kicked the conversion and a straight forward penalty seven minutes later.
A Laney penalty tied the scores with 14 minutes remaining, but Elwood had the last word as Connacht rekindled memories of Ireland's mauling of the Scottish national side last month.
Edinburgh: Laney, Webster, Di Rollo, MacDougall, Joiner, Godman, Blair, Jacobsen, Hall, Smith, Hines, Kellock, Strokosch, S. Cross, Blackadder.
Replacements: Kelly, Dickinson, Pringle, Callam, Lawson, Boston, Pyke.
Connacht: Mostyn, McPhillips, Yapp, McHugh, Slemen, Elwood, Keane, McFarland, Fogerty, Knoop, Short, Farley, Swift, Lacey, J. O'Sullivan.
Replacements: Jackman, Clarke, Buckley, Muldoon, O'Connor, Walls, Robinson.
Referee: Hugh Watkins (Wales)