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This was the Scottish side’s first away win since last April and they looked consigned to another defeat when they trailed 9-0 after 20 minutes thanks to three Paul Warwick penalties.
However tries from Marcus Di Rollo and skipper Todd Blackadder had got the visitors back into the contest with Chris Paterson’s two conversions leaving them trailing 17-14 at the break.
Connacht had increased their lead when South African prop Pieter Myburgh bundled over in the corner. And after Blackadder’s try had levelled the game Mark McHugh gave the Irish side a slim interval lead.
However Edinburgh took charge in the second period with prop Allan Jacobsen giving them the lead for the first time – and they never looked back. Laney added their fourth score soon after and then set up Simon Webster for the fifth as the Edinburgh side took control. Paterson completed his fifth successful conversion to open a 21 point lead and although Matt Lacey and Darren Yapp both scored late on there was no way back for Connacht.