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The Irish province opened the scoring when from a scrum scrum-half Chris Keane made a break into the Edinburtgh 22 where the home side conceded the penalty.
Elwood stepped up to slot the three points.
Edinburgh rallied with first centre Ben McDougall, then Marcus Di Rollo and Brendan Laney testing Connacht´s defence.
But they were caught with a sucker punch with number eight John O´Sullivan kicking ahead and left winger Dave Slemen winning the race for the touchdown. Elwood was once again on target with the conversion.
Again Edinburgh fought their way back into the game but a decision to opt for a scrum deep inside the Conacht 22 backfired as they lost possession.
Phil Godman missed a chance to claim three points before the break but failed, to leave Connacht 10-0 ahead at half-time.
The break did little for Edinburgh’s composure. A pass from Godman was soon intercepted, with Ally Kellock subsequently yellow-carded and Elwood kicking the penalty.
But Edinburgh rallied once more when Laney entered the line to start a long phase of passing and then set up Craig Smith to score. Godman kicked the conversion and a straight forward penalty seven minutes later.
With fourteen minutes left a Laney penalty tied the scores but Elwood ensured the Irish side went home with the spoils.
Final score: Edinburgh 13 (Smith try, Godman conv, pen, Laney pen), Connacht 16 (Slemen try, Elwood conv, 3 pen)