Boyle is a spoiler for Rose.
An injury-time strike by John Boyle set up a Scottish Cup fourth-round derby clash with rivals Cumnock.
Boss Sloan said: "To be honest I would have settled for penalties after we went down to 10 men, but the lads kept on going."
Bot opened the scoring midway through the first half when Mick McCann latched on to a pass from Boyle to fire home from close range.
Rose levelled on the half-hour as Mark Tyrrell spectacularly curled a 20-yard free-kick into the net.
The hosts were reduced to 10 men in the second half when Steven White collected his second yellow for a foul on Stuart McArthur.
Boyle then struck the winner in the last minute when he reacted quickest to bundle the ball in off the post.
Dario Quinn was sent off in the dying seconds as the holders exited.
Rose boss Davie Baikie said: "They did the same last week but this time their late goal put them through." TAYPORT ...0 PETERSHILL ...0 PORT forced a replay despite playing the whole game with 10 men after Derek Wemyss' first-minute red card.
Peasy's Kenny Hadden saw his free-kick hit a post as Tayport restricted the visitors to long-range chances.
The home side could have won it late on when sub Lee Bryce broke clear but his shot was saved by the keeper.
BANCHORY ..0 VALE OF LEVEN ..2 TWO early strikes saw Vale dump the home side to reach the fourth round.
Kenny Wilson fired the West of Scotland outfit ahead in four minutes and David McNaught added a second two minutes later.
Banchory battled bravely but despite creating a string of chances they failed to breach a resolute Vale defence.
LOCHEE UTD ..2 RENFREW ..3 SUPER-SUB Matt McLean's late strike sent Frew into round four.
Colin Forbes and Clinton Maxwell put the visitors two in front before Craig Robertson and Paul Blackwood drew Lochee level.
The home side were caught out right at the death when McLean broke clear to slot home the winner.
WHITLETTS .....1 KILBIRNIE .......1 A LATE strike from Alan Kinney salvaged a replay for Kilbirnie.
Ricky Hanvey fired home from the spot to put Whitletts ahead in the opening stages.
Blasties saw their cup hopes fading but right at the death Ryan McGregor was pulled down inside the area and Kinney slotted home the penalty.
Head boy: John Boyle parties