The Blues will take themselves out of that equation if they win this season's Heineken Cup.
Connor put the Dragons ahead with an early penalty at the Liberty Stadium, Dan Biggar - playing at fly-half with Hook in the centre - levelling matters with a drop goal for the home side.
Alun Wyn Jones put in a huge hit on Joe Bearman as the derby began to heat up, before Connor landed his second penalty.
We want to go to Munster in the last game and prove a point
Ospreys coach Sean Holley
Biggar's restart was plucked superbly out of the air by Bowe, the Lions tourist off-loading behind his back to Hook who handed off Dan Lydiate as he crossed for the opening try.
Hook missed the conversion, but extended his side's lead with a fine 25th-minute penalty.
Connor missed with a shot at goal, then saw his team down to 14 men when flanker Jamie Ringer was sin-binned for a clumsy challenge on Lee Byrne when the full-back jumped in the air for a high ball.
It was the Dragons who profited in Ringer's absence, though, Connor landing two penalties after Hook had missed an Ospreys kick to give the visitors an unlikely 12-11 advantage at the break.
The Ospreys were forced to rely on moments of individual brilliance
Following his knock Byrne was replaced as a precaution at half-time, the Ospreys then missing an early second-half chance to regain the lead when Hook blew another shot at goal.
Andrew Bishop stirred the home side with a fine midfield break, resulting in the Dragons again being reduced to 14 men as Rory Sidey saw yellow for killing the ball at the breakdown.
Hook landed the resultant penalty to edge the Ospreys back ahead, but the advantage was short-lived as Connor converted his fifth kick at goal.
A piece of Williams magic was needed, and the Lions winger dutifully obliged, a trademark midfield step breezing him past Rhodri Gomer Davies for a try under the posts.
The Dragons hit straight back, a Connor break and off-load splitting the Ospreys defence and forcing Bishop into an off-the-ball tackle on the supporting runners to prevent a try.
The centre was sin-binned, Connor kicking the resultant penalty but missing with a drop-goal that would have brought the scores level.
We went for the bonus point and unfortunately we didn't get it so we're pretty disappointed
Man-of-the-match Jason Tovey
Another glorious Williams break split the visitors' defence, Aled Brew killing the attack illegally to gift Hook a penalty under the posts, the Ospreys showing no ambition to go for the tries that would keep them in the title race.
The Dragons were still hopeful after Hook failed for a fifth time with a penalty, and the visitors nearly scored following an excellent Brew break.
A Hook drop goal with four minutes to go put the result - and the title race - beyond doubt.
Ospreys coach Sean Holley: "Very disappointing. It would have been tall order - we needed three bonus point wins, but just happy to get a win in the end, really.
Ryan Jones speaks to Scrum V
"We are trying to build some confidence, some momentum towards the end of the season.
"No [I wasn't disappointed we didn't go for the bonus-point try at the end of the game instead of the drop-goal]. We needed to win the game first.
"It clearly wasn't a night for tries and the Dragons made it really, really difficult for us and as did the referee.
"But we got the win and we'll look to get two more wins to finish the season of strongly.
"There's still [Wales] tour places to play for. We want to finish as high up in the league as we can.
"We want to go to Munster in the last game and prove a point and we'll just keep working because there's another season to play for."
Man-of-the-match Jason Tovey: "We came here to win, but when we knew the game was away from us we went for the bonus point and unfortunately we didn't get it so we're pretty disappointed about that.
"We're concentrating on getting up the league, getting into that European play-off and I hope to help the Dragons do that and if the Welsh tour [to North America] did come my way, I'd take it."
Ospreys: Byrne, Bowe, Hook, A. Bishop, Williams, Biggar, M. Phillips, D. Jones, Bennett, A. Jones, Levi, Wyn Jones, Smith, Holah, R. Jones. Replacements: Owen for Byrne (41), Spratt for Biggar (65), Knoyle for M. Phillips (80), James for D. Jones (51), Shervington for Bennett (41), Gough for Levi (51), Tiatia for R. Jones (65).
Dragons: Tovey, Wyatt, Gomer-Davies, Sidey, Brew, Connor, W. Evans, Black, S. Jones, R. Thomas, MacDonald, Charteris, Lydiate, Ringer, Bearman. Replacements: Walker for W. Evans (55), Gustafson for Black (59), Willis for S. Jones (59), L. Evans for Ringer (59). Not Used: N. Hall, A. Hall, Dolman.
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