Exteberria Benat's free-kick put Betis in front but Alvaro Negredo equalised before half-time.
Fazio was given his marching orders with just over 20 minutes remaining but the 10 men managed to keep their opponents at bay in the closing stages.
Negredo's early effort was tipped over by keeper Espinosa Casto and Jesus Navas then shot wide.
Benat and Jefferson Montero were off target for Betis but the former made the breakthrough in the 24th minute. Fazio fouled Ruben Castro and was booked, and further punishment followed when Benat curled the free-kick into the bottom corner.
Jose Antonio Reyes headed Navas' cross wide and Antonio Luna forced a good diving save from Casto as Sevilla pressed.
Casto then tipped over Fernando Navarro's header from a Piotr Trochowski corner but the equaliser finally came five minutes before half-time. Navas got to the byline and crossed from the right and Negredo headed past Casto.
Montero shot over before half-time and Sevilla keeper Javi Varas was into the action for the first time shortly after the restart when he parried Ruben Castro's shot and Montero put the rebound over.
Fazio blocked Juanma's close-range shot and Emir Spahic denied Benat in similar fashion.
Jose Canas steered Benat's corner wide of the far post before Fazio was shown a straight red card for fouling Roque Santa Cruz as he ran clear on goal. Benat's free-kick flashed across goal off a Sevilla head and narrowly eluded Santa Cruz, sliding in at the far post.
Undeterred by their numerical disadvantage, Sevilla pressed forward and Martin Caceres headed Ivan Rakitic's free-kick over the bar.
But play swung back to the other end and Javi Varas held another Benat free-kick and Spahic blocked efforts from Benat and Ruben Castro as Sevilla held on for a point.