Things didn't "continue to work" — major systems were unresponsive, offline, plain broken, or inscrutable when Starfleet arrived
From the first episode, "Emissary":
O'BRIEN: I'd like to ask the designer what he was thinking about when they built this place. I still haven't been able to find an ODN access. That's the Prefect's office up there.
SISKO: So all others have to look up with respect. Cardassian architecture.
O'BRIEN: Yes, sir. Major Kira's been using it.
SISKO: Is it my imagination or is it unusually warm?
O'BRIEN: The environmental controls in Ops are stuck at thirty-two C. We're working on it.
Also:
JAKE: Is this the food replicator?
O'BRIEN: I'm afraid they're all offline. There's plenty of emergency rations. I could send some down.
Later, O'Brien, a veritable transporter wizard, has trouble operating the transporter!
O'BRIEN: Trying to lock on. I've never done this with a Cardassian transporter. Damn it, what's the problem??
(He kicks the thing and Odo beams in.)
Starfleet installed their own components for many systems, because the Cardassians removed "every component of value" before they left
From the same episode:
O'BRIEN: We've got all available personnel assigned to repairing primary systems, sir. The Cardassians took every component of value. We're virtually defenceless.
Finally, in getting things to work and in getting used to the Cardassian systems, we mustn't underestimate the value of:
- universal translators
- previous Starfleet espionage of Cardassian technology
- investigation and reverse-engineering of Cardassian technology recovered from wreckage and/or captured ships during the Federation-Cardassian war
- assistance from Bajoran militia officers who lived under Cardassian rule
