There's two ways this can happen, I'd venture.
If you ask for a routing, that takes you through airspace, and the routing is granted, I'd take that as permission.
If, on the other hand, you simply inform an air-trafficker of your route, and have only asked for a service (RIS/FIR/RAS), then no permission has been granted, and therefore I'd venture that you need crossing clearance separately.
I can recall an occasion when I fell foul of this, I was working service A, with a RIS, who had my height and routing. They eventually handed me over to service B, whose first call after identifying me was to inform me that I was already 10 miles or 1000ft (depending upon how you look at it) inside their class D airspace without permission. I apologised, and they clearly had better things to do than worry about it - but I was almost certainly at fault since no specific clearance had been requested or given.
In an ideal world, ATC will identify that you will need a clearance, and sort it out before you need to ask for it. But, they are busy folks, and you can't assume that'll happen.
G