"Yesterday (Saturday), the death toll was at 10 people and now we are sitting at 15 as of this morning," Joe Phaahla told reporters at Tambo Memorial Hospital.
The tanker, transporting liquefied petroleum gas (LPG), was caught beneath a bridge close to the hospital and houses on Saturday morning in Boksburg, east of Johannesburg.
The minister said three hospital employees, two nurses and a driver, died later from severe burn injuries.
Thirty-seven people were injured, including 24 patients and 13 staff members who were in the hospital's accident and emergency unit at the time of the blast.
They "sustained severe burns and have been diverted to neighbouring hospitals", Phaahla said.
Others were hit by shattered glass, he added, while some were hurt as they were in the parking lot or in front of the hospital.
Videos on social media showed a huge fireball under the bridge, which the tanker appeared to have been too high to go under.
It was carrying 60,000 litres of LPG gas, which is used especially in cooking and gas stoves, and had come from the southeast of the country.
The health minister said the blast severely damaged the hospital's accident and emergency unit and X-ray departments, adding the roof was also damaged.