Rovers dominated the first half and led when Christopher Samba headed home from Morten Gamst Pedersen's corner.
Benni McCarthy's penalty, awarded after Ibrahima Sonko fouled Morten Gamst Pedersen, opened the scoring in the ninth minute.
Liverpool's wide players, Yossi Benayoun and Ryan Babel, also found the going tough - as did Blackburn's wingers Morten Gamst Pedersen and Bentley.
Yet there were promising early signs from the hosts, with Manuel da Costa seeing a towering header cleared off the line by Morten Gamst Pedersen.
Five minutes later Bentley was again involved when Benni McCarthy half-volleyed high and wide after another enticing cross that Mortem Gamst Pedersen did well to control.
Carrick diverted a Morten Gamst Pedersen cross towards his own net and, despite keeper Edwin van der Sar saving, Derbyshire sidefooted into the roof of the net.
Soon afterwards, Givet gave Rovers the lead with a stunning strike - smashing the ball into the corner of the net after Ciaran Clark had half-cleared Morten Gamst Pedersen's throw.
It was a well-constructed goal that saw Morten Gamst Pedersen play in Pascal Chimbonda, who in turn drilled the ball low across the face of goal for Kalinic to slot home.
English-based stars John Carew and Morten Gamst Pedersen will be well known to Scotland fans as will former Liverpool defender John Arne Riise, now at Roma, and ex-Tottenham forward Steffen Iversen.
City centre-back Kolo Toure left Morten Gamst Pedersen's hopeful punt upfield to Hart but the keeper -- having come 30-yards out of his goal -- missed the ball and Kalinic steered in a left-foot shot.
Everton keeper Tim Howard saved well from Morten Gamst Pedersen and Santa Cruz - a continuing transfer target for Manchester City and - twice headed off target from good positions following dangerous crosses by Steven Reid.
Blackburn almost scored with their first real opening against the run of play when Morten Gamst Pedersen's chipped cross presented Franco Di Santo with the simplest of headers, but the former Chelsea striker cleared the bar from six yards out.
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