Bellerin, who joined Betis on loan from Arsenal during the close season, looked to have put his side ahead in the first half when he latched on to a through ball and poked the ball between the legs of Sevilla keeper Bono and into the net.
But the goal was ruled out for a narrow offside and the home side were dealt another blow on the stroke of halftime when Guido Rodriguez was sent off for a second booking.
Sevilla took full advantage after the interval, Argentine fullback Marcos Acuna giving them the lead in the 55th minute with a thumping shot from outside the box following a layoff from compatriot Lucas Ocampos.
The visitors clinched victory on enemy territory when Bellerin attempted to clear a cross from Gonzalo Montiel and instead sent the ball into his own net before putting his hands to his head in disappointment.
The win took Sevilla up to third place in the standings on 27 points, level with second-placed Real Madrid and one behind leaders Real Sociedad, who won 2-0 away to Osasuna earlier on. Betis were left fifth on 21 points.