Borders had looked the keener of the two sides in the opening moments of the game but they could not turn pressure into points, and that failing was to prove critical.
They should have broken through when Ally Dickson was set free following a powerful pack drive, but he became isolated from support and the danger was dissipated by the Irishmen.
Connacht then seized the initiative and surged into a 10-0 lead.
They refused to let Paul Warwick's penalty miss affect their momentum and he made amends by scoring the opening touchdown following a break by David Slemen and skilful recycling work.
Warwick added the conversion before a poor tackle attempt from Stuart Moffat allowed Darren Yapp to embark on a 50-metre run upfield.
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 the Irishmen.
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 men 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.
Both sides were becoming increasingly frustrated and 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.
As the seconds ticked away, Connacht's confidence grew - and they effectively tied up the contest with a Mark McHugh penalty after another offence by McEntee.
|The Borders Score Card|
|Connacht Score Card|