Even though the address format is different (the prefix part) the transaction is valid so there will be a transaction on Dogecoin blockchain to the address.
I found a reddit thread where they suggest to save your private key in the Bitcoin client and import it to your Dogecoin client, although it is not certain this will work, I think this is your best chance, as transactions are irreversible by design.
In the original Bitcoin Core client you export in the debug window (based on this and this):
click on 'help' in the menu bar (top right)
click on 'debug window'
select the 'console' tab type: walletpassphrase "passphrase here" 600
type: dumpprivkey [address here]
Then to import navigate to the same window in Dogecoin client and
- type
importprivkey [private key here]