Brian Barry-Murphy's side cruised into a four-goal lead after a penalty from Oscar Bobb, a quickfire double from Kayky and a second-half strike from James McAtee looked to have set up a comfortable afternoon.
Late goals from Adam Wharton, Connor McBride and an own-goal from Finley Burns gave the hosts hope of an unlikely comeback, but substitute Dire Mebude sealed the win with a late fifth in injury time.
It was a steady start to proceedings in Leyland, neither side able to provide an early breakthrough.
However, City were handed a golden opportunity to take the lead when the referee awarded a spot-kick just after the 20th minute, with Bobb making no mistake to slot home past Rovers keeper Aidan Dowling.
And Brian Barry-Murphy's side weren't finished there, as Kayky added a quick-fire double before the break, latching onto assists from McAtee and Luke Bolton in the space of just three first half minutes.
We started the second period in a similar vein and added a fourth early on, this time with substitute Will Dickson laying on for McAtee.
It looked as though it would be a comfortable afternoon in the end for the EDS, that was until Wharton pulled one back for the hosts with 15 to play.
McBride then added a second just five minutes later before an own-goal from Burns set up a nervy final ten minutes for City.
However, Barry-Murphy's side dug deep to preserve the win, and would seal all three points in the final seconds as Mebude grabbed number five in stoppage time.
HOW WE LINED UP
Bobb was named in a midfield three alongside Romeo Lavia and McAtee, who were both fresh off the back of facing Swindon Town in the FA Cup.
Kayky, who also featured for the first team, was named on the right, with Bolton and Carlos Borges lining up alongside the Brazilian in attack.
WHAT IT MEANS
Victory sees City climb to second place in Premier League 2, just one point behind leaders West Ham United.
However, the Hammers and Tottenham Hotspur, who are directly below us, have two games in hand on us in the race for the title.
Next up, the EDS will take on Manchester United in Premier League 2 at the Academy Stadium.
Kick-off is at 19:30 (UK) on Saturday 15 January, with the game available to watch live on CITY+.
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