I do not find any reference to a Klingon cloaking device in TAS. I believe you might be thinking of the Stasis Field Generator weapon the Klingon ships used in "More Troubles, More Tribbles"
We may not have seen it used but the Klingons did have cloaking technology or mentioned having it in TAS
JB
Yes, that would be Spock's line of Dialog in The Time Trap:
Chakoteya.net said:KIRK: Explanation?
SPOCK: I can offer none at this time. However, the Klingon ship was not destroyed by our phaser fire. Its shields clearly deflected the phasers. Nor was its disappearance the type that would have been affected if it were using its cloaking device.
Kor