A try from Samoan flanker Ray Ofisa and three penalty goals from Ian Keatley saw Connacht cruise to a 14-0 half-time advantage.
The Blues lost hooker Rhys Thomas to the sin-bin just before the break but got back in touch thanks to two penalties from Ben Blair.
The New Zealander notched a third but Connacht regained control in the final quarter and looked set to end the game on the attack.
But the home side had an uncharacteristic lapse in defence, two minutes from time, and that was all Cardiff needed as a 35-metre break was finished off by Welsh international Sowden-Taylor.
Blair added the conversion to move the Blues ahead of the first time and rubbed salt into Connacht's wounds with an injury-time penalty.
Dai Young's men, minus eight international players, made some early headway and pinned Connacht back into their own half.
Xavier Rush, Sowden-Taylor and Scott Morgan carried well but the hosts soon relieved the pressure and Keatley popped over his opening penalty after 13 minutes.
Keith Matthews, who penned a three-year contract extension with Connacht this week, was the pick of the home backs as they began to put their imprint on the game.
Cardiff were caught out on 22 minutes, falling asleep as Connacht took a quick tap penalty and good hands out wide from the excellent Andrew Browne helped Ofisa through on an overlap for his try.
Keatley failed to convert but he was successful with two penalty attempts before half-time, following up on good work in the loose from John Muldoon, Browne and Robbie Morris.
Cardiff survived the loss of Thomas as Connacht, searching for their first win in four league games, began the second period tentatively.
Blair punished two early indiscretions - Connacht scrum half Frank Murphy was at fault for the second - to close the gap to 14-6.
Although they allowed Blair to boot the Blues to within five points, Connacht tightened up their discipline for the closing 20 minutes.
Helped by the return from injury of Ireland flanker Johnny O'Connor, Michael Bradley's side looked capable of holding on and gaining some much-needed ground in the table.
However, Sowden-Taylor and Blair combined in the dying moments to deny Connacht and notch Cardiff's fourth straight win in all competitions.