The programme sees entrepreneurs pitch for investment in the Den from the Dragons - five venture capitalists willing to invest their own money in exchange for equity.
BBC: Duncan Bannatyne: Dragons' Den star 'suffers heart scare'
But almost immediately he made a hash of a clearance kick from his 22, gifting the Dragons a five-metre line-out as the game entered its final play.
Scrum-half Thomas Bosc's eight kicks sealed the win to move the Dragons to within five points of top side Leeds.
Carey, who took charge at the Racecourse when Denis Smith was sacked in January, saved Wrexham from relegation after the Dragons secured four victories from the final five matches of last season.
BBC: SPORT | Football | My Club | Wrexham | Carey job under threat - Dickens
That four-point gap could have been brought back to one when the Dragons were caught offside five metres from their own line, but the Ospreys opted for a scrum instead and wasted their chance.
With the battle-weary Dragons having lost five in a row and Connacht having little to play for, the slow start at Rodney Parade could have been predicted.
Adam Jones was thwarted after the Dragons had set up a five-metre line-out and with both Frankie Sheahan and Gethin Robinson sin-binned for separate incidents, the game lulled.
The visitors had given up a penalty in the build-up and with Thomas back on the field the Blues opted to take a five-metre scrum, only for the Dragons pack to front up and disrupt their opponents.
Ronan O'Gara kicked over the first points five minutes from the break but James Arlidge replied for the Dragons to make it 3-3 at half-time.
Both teams continued to play open rugby after the interval and the Dragons wrested the lead from their opponents thanks to second-row Olivier Elima who ran over five minutes after the restart.
Bath's victory was vital to their Heineken Cup aspirations after Pool Five rivals Toulouse claimed a bonus-point win over the Dragons on Saturday.
Connacht were fresh from their opening-day win over Newport Gwent Dragons and made the more impressive start, with penalties from Ian Keatley on five and 10 minutes putting them 6-0 ahead.
But the Dragons switched back on and Sweeney's late drop-goal was enough to secure five points.
The pair held face to face talks with five players and are considering several other options as they look to strengthen the Dragons squad for next season.
Full-back Murphy glided inside Hughes and Tovey for a converted try eight minutes from time and an overthrown Dragons' lineout, five metres from their line, allowed Quinlan to close out the scoring.
The Reds reached the Super League play-offs for the first time last season under Karl Harrison but his five-year tenure was ended after their 66-6 loss away to the Catalans Dragons in May.
By now, Newcastle were rampant and Dragons were reduced to 13 men when Andrew Hall became the second player to be yellow-carded inside five minutes, also for preventing release.
Randall added another five minutes from half-time, linking up wonderfully with Purdham to pick a hole on the Dragons right.
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