Sounds like the tank prob is taken care of..
paper towels, hair dryer, fan...
Anything to push air over & around the tank (fan preferably).
I have pergot in my kitchen where one tank resides... pergot can take a good bit of water on it & drying, the problem more comes in when it seeps around to the un-butted portions ( i.e. drywall areas) usually it locks in fairly tight and creates a seal in the traffic areas.
That said, get as much up via paper towels, use a fan to maximize air flow and hope..
(I've spilt gallons and gallons of water on the finish part of traffic areas, mopping it up with a swipe or two casually in frustration, and then just had to swifter/wipe the floors due to residue later...if it's on the finish part, you are ok unless you leave large amouts of standing water)
Edit - Forgot to mention, the particle board nature of the pergot will sort of resist rotting if the water isn't a huge amount.. I assume you tank is 4' or less, asuming a standard size of sorts..
The standing water of that amount wouldn't cause dry rot (prolly only just crack the finish at worst).
For reference, I went on vacation and had the dishwasher hose burst and leak about 40sqft of water on to my pergot while I was gone (the basement was a nice waterfall).
End of the whole ordeal only the area directly around the dishwasher even cracked, no real structural damage, mold etc to either the pergot or the subfloor. (prolly a ft diameter from the appliance).
Oh, and it's fairly easy to spot replace pergot, I'd be happy to PM you how if it comes to that.