Another Feret set-piece, this time a free-kick, created a headed chance for substitute Victor Montano but the forward could not keep his effort down as he rose unchallenged in the box.
Shearer rose above Gianluca Festa in the penalty box to carefully place a header into the left-hand corner of the Middlesbrough net, giving Schwarzer little chance.
But Bothroyd's second on the hour - his fourth of the season - put Cardiff in control as he rose unmarked in the six-yard box to nod in substitute Burke's right-wing cross.
Five minutes before half-time Rose made amends, finding space in the box to side-foot home after Carroll had nodded down a deep ball from Michael Mancienne.
The cross-platform nature of Adobe AIR means the iPlayer will work with Open Source and Apple Mac computers "out of the box" on 18 December, said Mr Rose.