The victory was the Ospreys' first away from home in the league since the 2006/7 season, a welcome fillip for new coach Sean Holley.
Connacht, with four new recruits in their line-up including half-backs Keatley and Frank Murphy, had failed to cope with facing the wind in the opening half.
Despite the 13-0 deficit, hopes were raised when visiting hooker Huw Bennett saw yellow early in the second period.
But the Ospreys had enough nous to hold out, with former All Black flanker Marty Holah showing his streetwise skills.
Teenage scrum-half Rhys Webb was undaunted amidst the Ospreys' 13 internationals and the Wales Under-20 star showed that Jamie Nutbrown will have to fight for the number nine jersey when he arrives from New Zealand.
Connacht's frustration grew, as did their penalty count, while Keatley missed a 67th-minute shot at goal.
Connacht lost Andrew Farley to the sin-bin for the closing stages, and they had to be on their toes in deepest injury time to prevent debut-making Ireland international Tommy Bowe from scoring a breakaway try.
Ospreys coach Sean Holley: "The conditions were atrocious, so it was all about being focused, accurate and solid in defence.
"We got the early lead, but it was important not to concede when we weathered the Connacht storm early in the second half.
"We came through that, grew in confidence and deserved the win."
Connacht: Carr, Bibo, Duffy, Matthews, Hearty, Keatley, Murphy, Wilkinson, Cronin, Morris, McCarthy, Farley, J. Muldoon, Ofisa, Rigney. Replacements: Nathan for Keatley (76), Flavin for Cronin (70), Loughney for Morris (70), Swift for Rigney (74). Not Used: Browne, Campbell, Wynne.
Ospreys: Byrne, Walker, Parker, A. Bishop, Bowe, Hook, Webb, James, Bennett, Griffiths, Gough, Evans, Lloyd, Holah, R. Jones. Replacements: Wells for Webb (68), D. Jones for James (70), Tiatia for Bennett (57), Shervington for Lloyd (49). Not Used: Wyn-Jones, Owen, Williams.
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