It took the hosts just six minutes to break the deadlock when Alvaro Medran broke through they away side's rearguard to fire in his sixth goal of the campaign.
However, visiting Conquense were level in the 17th minute when Oscar Vega stole in to convert from close range.
It appeared that the spoils were going to be shared, but in the 65th minute Burgui struck his second goal of the season for Castilla, which sealed all three points in the process.
The match also saw the Castilla debut of Enzo Zidane, son of Real Madrid legend Zinedine.
The victory has lifted Castilla up to sixth in the Segunda B standings.