The Scottish basement boys had taken on the tag of favourites against a team that had lost five in a row.
But they were unable to reproduce the form that had earned them positive results against Leinster and Edinburgh.
Borders had looked the hungrier outfit in the opening sparring session, but they could not turn pressure into confidence-boosting points, and that failing was to prove crucial.
They should have broken through when Ally Dickson was released in the wake of a powerful pack drive, but he became isolated from support and the danger was mopped up.
Having weathered the early storm, it was Connacht who seized the initiative by surging into a 10-0 lead.
They refused to let Paul Warwick’s penalty miss affect their momentum and he made amends by bagging the opening touchdown following a break by David Slemen and slick recycling work. Warwick added the conversion before a poor tackle attempt from Stuart Moffat allowed Darren Yapp to embark on a 50-metre dash.
He was eventually brought down, but they maintained their patience to create the time and space for Slemen to slot a tidy drop-goal. Ally Warnock eased the Borders’ jitters with a penalty, but the rest of the half belonged to Connacht. They remained camped inside the home half for 15 minutes but they could only manufacture one try during that period of dominance.
The latest in a long series of forward drives paved the way for hooker John Fogerty to plunge over.
Borders coach Steve Bates would have had plenty to say to his troops during the interval and they came back out looking meaner and more alert - but they were up against a well-drilled and resolute Connacht defence. They showed no real signs of making a significant breakthrough, though they did whittle away at the visitors’ lead with two more Warnock penalties.
Welsh referee Tim Hayes marked his league debut by sin-binning rival breakaway forwards Wayne McEntee and John O’Sullivan.
Borders survived their spell without the former, but Warnock was wide with another penalty bid when the latter was shown the card.
Connacht’s confidence grew - and they effectively tied up the contest with a Mark McHugh penalty after another offence by McEntee.
: Pens: Warnock 3.
: Morton, Moffat, MacRae, Dickson, Walker, Warnock, Weston, Thomson, Ford, Cross, Blair, Stewart, Dalziel, Millar, McEntee. Replacements: McGee for Cross (60), Brown for Blair (49).
: Tries: Warwick, Fogerty. Cons: Warwick. Pens: McHugh. Drop Goals: Slemen.
: Mostyn, McPhillips, Warwick, McHugh, Yapp, Slemen, Tierney, Hogan, Fogerty, Bracken, Short, Farley, Swift, Muldoon, J. O’Sullivan. Replacements: Keane for Tierney (49), McFarland for Hogan (58), Knoop for Bracken (45), O’Connor for J. O’Sullivan (79).
: T Hayes (Wales).
: Cardiff Blues 12 Edinburgh 17
: Saracens 32 Newcastle 13; Leeds 16 London Irish 5.