Yes, you 'put the key into the lock'. Both 'put' and 'place' means to place something in a specified location or condition. 'put into' means more to attach ,tack or append ,which is the case with attaching the key in the lock. It is more suitable to use 'put' here.

'put" is a more general expression than "place". You can replace "place" with "put", but not always the other way round.