Ten unanswered points in the second half allowed the 14-man Borders to beat Connacht for their first away win in the Celtic League.
Connacht: (7) 15:
Tries: Rigney, Lacey
Borders: (7) 17:
Tries: Hore, Danielli
Cons: Hore 2
The Irish side were 15-7 ahead at the break with Colm Rigney and Mark Lacey touching down after Semo Sititi was red-carded in the 10th minute.
Borders fly-half Chris Hore had converted his own first-half try to initially help the Borders to lead.
Simon Danielli scored on the restart with Hore kicking the winning penalty.
Connacht are now struggling in the basement of the league after the Borders finally won an away game after 52 games in the competition.
Paul Warwick booted Connacht into an eighth-minute lead from a close range penalty.
Two minutes later, Sititi was sent to an early shower as at a set scrum, he swung his arm wildly at the face of Connacht's Tom Tierney and Welsh referee Nigel Owens had little option but to reach for his red card.
An injury to Warwick did little for Connacht's cause as did Hore's 21st-minute try. The Kiwi fly-half blazed through under the posts and added the extras for 8-7.
Connacht struck back on 27 minutes when flanker Matt Lacey muscled over to score from a lineout maul.
Connacht scored again after lock Mark Blair was sin-binned for a ruck infringement in first half injury-time and with Colm Rigney trundling over at a scrum with David Slemen converting.
A Stuart Moffat incision on 55 minutes sent Simon Danielli haring over for a try in the right corner.
Hore brilliantly drilled the conversion over and after Connacht flanker Michael Swift was yellow-carded, the Borders number 10 held his nerve to condemn Connacht to their third straight defeat.
Connacht: Mostyn; McPhillips, Yapp, Downey, Matthews; Warwick, Tierney; McFarland, Fogerty, Knoop, Short, Farley, Swift, Lacey, Rigney.
Replacements: Venter, Hogan, Muldoon, O'Connor, Keane, Slemen, Robinson.
Border Reivers: MacRae; Walker, MacDougall, Law, Danielli; Hore, Cusiter; Thomson, Scott, McGee, Palepoi, Blair, Dalziel, Sititi, Miller.
Replacements: Ford, Cross, Stewart, McEntee, Brown, McKearchar, Kibble.
Referee: Nigel Owens (Wales).