Nathan Dyer had given Swansea the lead before Michael Chopra's leveller just after the break.
Before Bothroyd's 77th-minute goal, Michael Chopra spurned a fine chance to get Cardiff back on level terms.
Peter Whittingham scored from the penalty spot four minutes in after Ben Turner held back Michael Chopra.
The result was perhaps a little harsh on the Bluebirds, who struck the woodwork twice through Michael Chopra.
And Keane altered his forward options with Wallace and Michael Chopra replacing Stokes and the energetic Daryl Murphy.
Jay Bothroyd hit the first of his two goals on two minutes, slamming home after Michael Chopra's shot was saved.
Similar to Michael Chopra's hero's return to Cardiff City, Trundle is back in South Wales seeking solace and the goalscoring touch.
Cardiff, bidding to become the first Welsh side to play in the Premier League, led twice through Michael Chopra and Joe Ledley.
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"I would like to get Michael Chopra back on loan, " admitted Jones.
The looser definition of Anglo-Indian (any mixed British-Indian parentage) encompasses the likes of cricketer Nasser Hussain, footballer Michael Chopra and actor Ben Kingsley.
John Parkin gave the Bluebirds an early lead with a stunner on his Cardiff debut as the striker smashed Michael Chopra's flick home from 25-yards.
The Black Cats then had a penalty appeal when Michael Chopra tangled with Alexandre Song Billong in the area but referee Keith Stroud was not convinced.
But in an incident-packed opening quarter, only a fine save from veteran Rovers stopper Neil Sullivan kept out Michael Chopra as the Cardiff striker pounced on Bothroyd's flick.
The visitors did go close early on when former Cardiff striker Michael Chopra fired goalwards from 12 yards, but his effort was blocked by Ian Evatt six minutes in.
Aaron Ramsey has left for Arsenal, Cameron Jerome to Birmingham, Chris Gunter to Tottenham, Michael Chopra to Sunderland - before his return last season - and Glenn Loovens to Celtic.
The Cardiff manager also said striker Michael Chopra suffered a stomach upset on the eve of the win over Preston and that players returning from injury showed character in difficult circumstances.
So Jones is keen to keep the former Arsenal trainee as although Cardiff boast a strike-force of Craig Bellamy, Michael Chopra and Andy Keogh, Bothroyd's role in his side is unique.
Burke, only recalled because of injury to Michael Chopra, had danced around the covering defenders before poking a miss-hit effort through a crowd of bodies and it trickled into the corner of Millwall stopper David Forde's net.
The Black Cats kept surging forward and arguably their best chance fell to substitute Michael Chopra with just minutes left on the clock, the striker failing to connect five yards out with Cerny grounded and the goal gaping.
But Ramsey's arrival adds further expectation for the Bluebirds boss to secure promotion to the top-flight this season as Ramsey would join a star-studded squad that includes Craig Bellamy, Jay Bothroyd, Michael Chopra, Peter Whittingham and Seyi Olofinjana.
New boss Mackay has only 15 recognised first-team players at Cardiff after Conway's arrival as 12 players left the club after last season's failed play-off campaign, including last season's top scorer Jay Bothroyd and fellow striker Michael Chopra.
Despite that strike rate McCormack, who signed a new four-year contract at the start of this month, says he fears he will not be first choice up front this season, with Bothroyd and Michael Chopra both also in form.
The lead took the sting out of the game and Cardiff tried to get themselves back into it with Ross McCormack, back for his first start in almost three months for suspended top-scorer Michael Chopra, lifting the ball over the bar from the edge of the box.
Michael Chopra and Joe Ledley twice put Cardiff ahead with crisp strikes but a Charlie Adam free-kick and a close-range finish from Gary Taylor-Fletcher drew the Lancashire side level on each occasion - before Brett Ormerod struck the winner shortly before the interval with a low strike.
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