Eric Elwood ended Connacht's two-game losing run with an injury-time penalty to defeat the Borders in Galway.
Connacht (10) 29:
Tries: O'Loughlin, O'Brien, Farley
Pens: Elwood 4
Border (22) 27:
Tries: Thomson, Brown, Hore 2.
Con: Hore 2
Borders led 22-3 after 30 minutes and scored four tries with Charlie Hore twice, Kelly Brown and Peter Thomson touching down.
But second half tries from Conor O'Loughlin, Niall O'Brien and Andrew Farley rescued the Irish side.
Connacht finally overpowered the Scots with Elwood contributing 14 points with his trusty right boot.
Borders open the scoring through 22-year-old blindside Brown who was bundled over in the right
Hore clipped the conversion over and added a penalty three minutes later
for a 10-0 lead.
Elwood, making his first start of the season, got Connacht back in the game
with a 27th-minute penalty.
But Michael Bradley's charges allowed the Borders the latitude to hit back straight away as two minutes later prop Thomson gleefully scored his second in
Hore then collected his own chip to steal over and convert for a 22-3 lead.
A sniping run from scrum-half O'Loughlin deservedly saw Connacht net their first try with five minutes of the half remaining, with Elwood converting for a
22-10 half-time deficit.
But Hore, dynamic in the loose, was put over in the 50th minute by fly-half
Alistair Warnock to confirm Borders' second bonus point of the season in the far
A Paul Warwick pop pass put Connacht winger O'Brien over seven minutes later and Elwood knocked over his third penalty.
Borders crumbled again as Farley lurched over, and although he missed the conversion his penalty in the ninth minute of injury time brought home the points.
Connacht: Connacht: Slemen, O'Brien, Warwick, Downey, McPhillips, Elwood,
O'Loughlin, McFarland, Fogerty, Bracken, Myburgh, Farley,
Neville, Lacey, J. O'Sullivan.
Replacements: Bourke, Clarke, Short, Muldoon, Walls, Robinson, Meagher.
Borders: Moffat, Walker, MacRae, Hore, Steel, Warnock, Weston,
Thomson, Ford, Douglas, MacLeod, Stewart, Brown, McEntee, Blair.
Replacements: Scott, Cross, McGee, Rennick, Millar, McKearchar,
Referee: Nigel Owens (WRFU).