In testing conditions the hopes of a flowing game were remote and in an error-strewn contest end one try proved crucial.
Connacht opened the scoring with a penalty from fly-half David Slemen as the home side were penalised for straying off-side.
The Reivers rallied to take the lead with two successful kicks of their own.
Calum MacRae slotted the first following another off-side offence before veteran Gregor Townsend slotted a drop goal.
However, the Irish province took control of the game once again when John Muldoon grabbed the only try of the game.
The flanker benefitted from a great challenge by Nikki Walker on Mark McHugh before Reivers' were dealt a blow with Semo Sititi sent to the bin.
Before the end of half Slemen and Reivers' rookie James King, on as a blood replacement for McRae, traded penalties
In a scoreless second half Borders were forced to reshuffle their line-up after the restart due to an injury sustained by winger Simon Danielli who crashed into an advertising board.
However they were unable to overturn the deficit and sustain their attempt to catch Scottish rivals Edinburgh in the table.
|Border Reivers Score Card|
|Connacht Rugby Score Card|