Sleepy Floyd and Buck Johnson added 18 points each.
Rivers's jumper made it 112-110 and he followed with a steal off Buck Johnson.
Houston had a chance to tie the score with a 3-point field goal with 14 seconds left, but Buck Johnson's shot went off the rim.
As he was about to shoot, Buck Johnson stepped into the lane and then stepped back.
Announced they will not extend a qualifying offer to Buck Johnson, forward.
Houston had pulled into its fifth tie of the fourth quarter at 102-102 with 15 seconds to play on a basket by Buck Johnson.
Robinson's scoring average was the highest at Alabama since Buck Johnson's 20.7 in 1986.
Buck Johnson tied his season-high with 19 points for Houston, which has won 10 of its last 12.
Buck Johnson scored 20 points for the Rockets and Mike Woodson added 15.
"Once, Buck Johnson tried to post me up and leaned on my right side - otherwise, there wasn't a lot of pain," he said.