The fly-half slotted an 82nd-minute penalty - his fifth of the game - to secure his side victory after the unheralded Irish province had threatened an upset.
Connacht had taken a 17-15 lead when centre Keith Matthews sped over for a try following a fine break from wing Conor McPhillips.
But referee Rob Dickson then pulled up the visitors for apparent time -wasting and Connor made no mistake with the straightforward penalty.
The Ospreys gained the win, but they were a shadow of the side who won the Celtic League so stylishly last season.
Connacht turned around with a 9-3 lead against last season’s Celtic League champions.
The Ospreys were beyond poor in the opening half in which they made numerous mistakes. Tackles were missed, passes went down, decision-making throughout the side was questionable and they were made to pay by the visitors.
Matthew Jones put the hosts into a 3-0 lead after just two minutes, but Connacht were not long in responding.
Paul Warwick slotted a penalty to reward some forward pressure and went on to complete a first-half hat-trick as the Irish province produced arguably their best half of rugby all season.
They looked on course for victory when Matthews touched down as the visitors took control at 17-15.
But they were penalised in the 82nd minute when the home side were awarded a penalty which Connor converted to earn a narrow win.
: Cashmore, Karauna, A Bishop, Davies, S Williams, M Jones, Spice, D Jones, B Williams, A Jones, Cockbain, Evans, Thomas, Tandy, Lloyd. Replacements: Connor for M Jones (40), Bennett for B Williams (57), Newman for Cockbain (56), Beech for Tandy (47).
: Mostyn, McPhillips, Hearty, Matthews, Yapp, Warwick, O’Loughlin, Hogan, Merrigan, Knoop, Short, Farley, Swift, O’Connor, Muldoon. Replacements: Riordan for Mostyn (72), Robinson for Hearty (52), Keane for O’Loughlin (65), Loughney for Hogan (67), Fogerty for Merrigan (55), Rigney for Swift (35), Lacey for O’Connor
: Rob Dickson (SRFU).
: Edinburgh Gunners 17 Newport Dragons 15.