I was charged with 3rd degree burglary and I was told that upon successful completion of my probation, that my charges would be dismissed.

I have been on probation for a year now and I have completed all that they required of me, however I was sentenced to a term of up to 5 years.
I would like to get off early so I can join the service, I have received confirmation that I would be allowed to join the service because my record will be dismissed.
If I petition to get off probation, would I be successful in Minnesota?

After one year? Personally I doubt it but you can try.

You should do at least half of your probation before asking for early termination in any state.