Evan Dirksen had a game-high 25 points, and Jack Armstrong added 15 more as the Chieftains won a non-league road game.
Alex Miller added 10 points for River, which led most of the first half before fending off a second-half comeback by the Thunder.
After River took a 40-35 lead into halftime, Mountain View rallied to take a 54-52 lead late in the third quarter.
River retook the lead as the fourth quarter opened. Mountain View got within 63-61 on a Kaden Horn 3-pointer with about two minutes to play before River sealed it.