Mike Penning said the incident happened at the Otterburn Training Area in Northumberland
The soldier was from the Royal Regiment of Scotland, and police and the Ministry of Defence (MoD) are investigating.
Mr Penning said: "My thoughts are with the soldier's family, friends and colleagues at this difficult time.
"The safety of our personnel is our absolute priority and, while deaths in training don't happen often, any death is a tragedy.
"As well as a police investigation, MoD accident investigators are looking into the circumstances surrounding this tragic incident."
The MoD said the soldier who died was male and he was shot during a training exercise on Monday night.
A spokesman said the Defence Safety Authority is investigating, but added that there were no details on whether any other personnel were involved.
The accident happened on a firing range at the training area, and Northumbria Police said the soldier died after suffering a "serious head wound".
A spokesman said: "At around 11.15pm last night police received a report that a soldier had been shot on the military ranges in Otterburn during a live-firing exercise.
"Emergency services attended and found the male soldier had received a serious head wound. Sadly he was pronounced dead at the scene by paramedics.
"Police will be working with the Ministry of Defence to establish the exact circumstances of the incident and a cordon is currently in place.
"Military officials have informed the soldier's next of kin."
The MoD has not yet released the soldier's identity.
First Minister Nicola Sturgeon said: "This is a tragic incident and my deepest condolences go to the family, friends and colleagues of the soldier who has died.
"Our armed forces do a difficult and dangerous job at all times, and this is another stark reminder of that. My thoughts are with all who loved this brave soldier."