Reivers made heavy work of their first away win from Netherdale after Samoan international Semo Sititi was shown a red card in the 10th-minute for a swinging arm.
However Reivers dug deep and were rewarded in the 21st minute when Hore found some space and sprinted through to score under the posts - making easy work of the conversion.
The hosts were not helped when goal kicker Paul Warwick was injured but they hit back through a Matt Lacey try. Warwick's replacement David Slemen missed with the conversion.
Connacht got their second try when Colm Rigney took advantage of Borders lock Mark Blair's yellow card and picked up from the back of the scrum to barge over. Slemen added the extras for a 15-7 lead at the break.
The second half sprung into life with 25 minutes remaining when Stuart Moffat carved up the defence to send Simon Danielli over for a try in the right corner.
Hore, who missed a penalty shot only minutes earlier, nailed the conversion and when flanker Michael Swift was sinbinned for coming over the top, he condemned Connacht to their third straight defeat.
|Connacht Rugby Score Card|
|Border Reivers Score Card|