Eagles' only defeat in six round-robin group matches came from the Bees, who recorded a 5-1 home win over the Grand Slam winners - and beat them to top spot in the table.
Although Ayr recognise any tie will be difficult, their 25-year-old Canadian forward Karry Biette admitted that Bracknell was probably the worst draw they could have had. "If you had to pick one it would be Bracknell because of what they did to us in the round-robin," he said.
"We did not fare too well in their rink but beat them at home. Bracknell have changed quite a lot and they seem to be gelling together early in the season. It will be a good battle."
The London Knights coach, Jim Fuyarchuk, who saw his men pick up their first wins against the Guildford Flames at the weekend, was relishing meeting Cardiff Devils.
"Cardiff look - as they have been in the past - really tough, really competitive," Fuyarchuk said. "There is no doubt we are going to have our hands full."
Newcastle Riverkings pair Blake Knox and Jim Mathieson face their former club, Nottingham Panthers.
In the other quarter-final, Group B winners Manchester Storm meet Sheffield Steelers.
Little separated the pair in the round-robin matches, one game ending in a draw while Storm nicked the other by one goal. A close tie is expected again.
BENSON AND HEDGES CUP Quarter-finals: Cardiff Devils v London Knights; Bracknell Bees v Ayr Scottish Eagles; Sheffield Steelers v Manchester Storm; Nottingham Panthers v Newcastle Riverkings.
BENSON AND HEDGES PLATE Quarter-finals: Edinburgh Capitals v Guildford Flames; Telford Tigers v Kingston Hawks; Fife Flyers v Peterborough Pirates; Paisley Pirates v Slough Jets.