Striker Ruben Castro scored twice and set up a third, becoming the joint-highest Spanish scorer in La Liga alongside Sevilla's Alvaro Negredo with 15 goals.
With eight games left, Betis are level on 47 points with fifth-placed Champions League quarter-finalists Malaga who they trail due to an inferior head-to-head record.
Betis's Dorlan Pabon raced clear two minutes later only to see his lob come back off the crossbar but Castro was on hand to put away a second goal for the visitors.
Granada were still reeling from that double blow when goalkeeper Tono raced out to meet a Betis free-kick but missed his punch and Paulao's looping header hit the bar.
Carlos Aranda's angled diving header in the 72nd helped Granada salvage some pride but they finished with 10 men after Guilherme Siqueira was shown a second yellow card in the 80th.
Betis's Angel Lopez completed the rout by running the length of the pitch to score on a counter-attack at the end.
Granada's first La Liga campaign in 35 years is heading for an uncomfortable finale. They are 16th with 28 points, having claimed only two from their last seven games, and sit just four points above the relegation zone.
