Yes, past time, even 1 minute, can call for past tense. We use past tense when we wish to indicate that the event is completely finished and behind us.
Present perfect also talks about past time, but the event is still in some way connected or related to the present time.
Often you can choose either form. In your sentences, both are OK:
I lost my key! (It is gone.)
I've lost my key! (and I don't know what to do)
They both express past events. The event occurring only one minute into the past may encourage a speaker to use present perfect form, but it is not necessary; past tense is OK too.
- For Teachers