Ryan O'Toole opened the scoring in style from 20 yards with ten minutes played, and Adam Martin doubled the lead when he headed in an Ollie Burgess corner.
Aylesbury hit back through Brian Haule, but Slough restored their two-goal lead as Burnell followed up to score after his own shot had been parried by Steve Smith.
The visitors struck again to reduce the arrears, with Lamar Simpson capitalising on poor defending, and Slough took a 3-2 lead into half time.
In the second period, the industrious Martin scored his second when he looped a header past Smith following another corner, but Aylesbury ensured a nervous final six minutes plus stoppage time when Luke Abraham scored with a fine 25-yard free kick.
Aylesbury Line up
Steve Smith, Max Farrelly, Luke Abraham, Davis Haule, Kevin Ayopolis, Liam Fowler, Jamie Connor, Lamar Simpson, Brian Haule (c), Sonny French, Linton Goss. Subs: Craig Faulconbridge, Jack Page, Stuart Bryant, Matarr Touray.