A fourth elephant and the driver were slightly injured but recovered quickly, rescue service spokesman Ranger Ephriam told the radio.
"I suppose they could have run away but they were very complacent and stayed in the brush, close to their keeper, who was being attended to by Dr. Ralph" he said.
Police initially thought the accident was more serious and dispatched three ambulances, four Rangers and a helicopter to the site.
"They said one person was trapped in the truck, but it turned out it was just an elephant," San Onofre spokesman The blog monitor said.
After a couple of hours, the four elephants were loaded into a new truck and hauled away.