The former FC Sydney star, who was signed last summer, rifled home a 42nd minute winner for Bryan Robson's Championship side.
And it was no more than the Blades deserved against a largely unfamiliar Wanderers line-up.
Trotters boss Gary Megson admitted that Premier League survival took precedence over the FA and UEFA Cups in the build-up to the match.
And that was underlined by his team selection; seven changes to the side that edged victory against Derby.
They included a home debut for keeper Ali Al-Habsi, plus first starts for new Israeli recruit Tamir Cohen and Swiss international Blerim Dzemaili.
Conspicuous by his absence was Nicolas Anelka, whose days at the Reebok Stadium are believed to be numbered.
Wanderers will announce Portsmouth midfielder Matt Taylor as another new recruit next week.
The fee for the former England Under-21 international is likely to be paid by the £10million-plus Wanderers receive for Anelka.
Bolton could have done with the goalscoring services of the former Arsenal, Liverpool, Real Madrid and Man City frontman.
They rarely tested Paddy Kenny in the Blades goal despite the need for an equaliser after Carney's first-half strike.
United fan Kevin Davies was suspended for this game and there will be a pressing need to provide him with a strike partner should Anelka depart.
In contrast, Robson can feel hopeful United can climb the Championship table.
He finished the tie with three out and out strikers in Jon Stead, fit again Rob Hulse and Billy Sharp.
The visitors also did without the services of Gary Speed, who was ineligible after his loan move from the Reebok.
United's winner came three minutes from the end of a dreadful opening period.
Stephen Quinn's through ball dissected the home rearguard and Carney - whose previous clubs included Everton, Oldham and Halifax - had time and space to finish crisply past Al-Habsi.
Bolton's only shot of note was an effort from Daniel Braaten blocked by Matt Kilgallon.
Wanderers made more of a first of things in the second half; introducing new recruit Dzemaiili at the break.
However, chances were still at a minimum with Danny Guthrie forcing Kenny into one decent save and Lubomir Michalik forcing the ex-Bury man to get his knees dirty.
The Blades didn't produce much of a cutting edge either and Al-Habsi was far busier on his previous appearances in the Carling and UEFA Cups.
However, with Bolton looking unlikely to force a replay, United didn't need to extend themselves looking for a further goal.