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Having gone in at the break 21-13 in arrears, Glasgow dug deep and managed to hit the front thanks to a Colin Shaw try and a Colin Gregor penalty, but the defending champions came good with a controlled finish.
Led out by hooker Barry Williams, who was making his 100th appearance for the region, the Ospreys hit the front as Connor picked off a well-taken drop goal before Glasgow had their fullback Bernardo Stortoni sin-binned for a deliberate knock on.
A Connor penalty kick followed and the Glasgow defence was then carved open on the 20-minute mark when Connor and the returning Gavin Henson, who was making his first start in eight months, combined to send fullback Byrne over for an unconverted try in the corner.
But errors started to creep into the Ospreys' game and Glasgow managed to tally up 13 points in a profitable nine-minute spell. Two Gregor penalty kicks sandwiched a brilliant try from scrum half Sam Pinder, whose pace helped him scurry over from 40 metres out.
Gregor's second penalty kick in the 30th-minute had Glasgow 13-11 ahead but their lead was a short-lived one as Byrne's second try, set up by a close range offload from Justin Marshall, and an injury-time drop goal from Connor restored the Ospreys' advantage for the interval.
The topsy turvy nature of the game continued on the resumption with Glasgow barging back into a 23-21 lead. Gregor kicked an initial penalty and winger Shaw showed a clear pair of heels to the home defence to touch down after a 20-metre dart.
The Scots were suddenly contemplating their first ever win at Liberty Stadium. Brave as they were, they were overhauled in the final quarter as the Ospreys found an extra gear to set up three penalty chances which Connor duly converted.
Armed with a 30-23 buffer, Lyn Jones's side made sure of the points in the final minute of normal time when replacement Lewis crossed the whitewash, much to the delight of the bulk of the 5,000-plus attendance.
Ospreys (21) 37
Tries: Byrne (2), Lewis
Cons: Connor (2)
Pens: Connor (4)
Drop-goals: Connor (2)
Glasgow (13) 23
Tries: Pinder, Shaw
Cons: Gregor (2)
Pens: Gregor (3)
Ospreys: Byrne; Terblanche, A. Bishop, Henson, Vaughton; Connor, Marshall; James, B. Williams, Griffiths, Cockbain, Lloyd, Smith, Tandy, Tiatia (Capt)
Replacements: Brew for Byrne (55), Owen for Connor (80), Roberts for Marshall (72), Millward for James (58), Hibbard for B. Williams (63), Bateman for Lloyd (60), Lewis for Smith (68)
Glasgow: Stortoni; Shaw, Morrison, Gibson, O'Hare; Gregor, Pinder; Collins, Milligan, M. Low, Newman, Kellock (capt), Swindall, Eddie, Beattie
Replacements: Barrow for Gibson (72), M. Evans for O'Hare (58), Kalman for Collins (40), McArthur for Milligan (68), Turner for Newman (68), Forrester for Swindall (77), Vernon for Eddie (55)
Att: 5,417
Ref: C Stanley (RFU)
