Xavier Henry had pulled the Grizzlies to within two at 106-104 on a basket with 5 seconds left. But Billups then canned a pair of foul shots after being fouled by O.J. Mayo, giving Denver a four-point edge.
Mike Conley of Memphis hit a 3-pointer at the buzzer to provide the final margin.
Nene contributed 27 points and grabbed 11 rebounds and Arron Afflalo pumped in 25 points in the Nuggets' seventh straight win.
Rudy Gay countered with 24 for the Grizzlies, losers of three straight.