Burglariously:

Not only is it a word, but the related “burglarious” has the Sir William Blackstone seal of approval. Westlaw also manages to beat the OED on this one; “burglariously” is attested by the OED only as far back as 1807, but a quickie Westlaw search locates a 1792 North Carolina case reporting an indictment “for feloniously and burglariously breaking and entering into the dwelling house of one Rice.” Google Books finds a reference from The Gentleman’s Magazine in 1789; Galenet finds one from 1724, which apparently paraphrases a 1640 statute as providing that

If any be Indicted or Appealed, for the Death of any evil disposed persons Attempting to Murther, Rob, or Burglariously or Feloniously to Break any Mansion House (and the same is so found by Verdict) he shall Forfeit no Lands or Goods for the same, but shall be fully Acquitted thereof ….

The oldest source I could find for “burglarious” is an index to a legal handbook published in 1749. (I’ll follow Sasha’s lead by reporting these early references to the Oxford people, since they try to include the earliest available references in their entries.)

I do not advise that you use the terms, though, except for humorous effect.