Ruben Castro scored both times that the sides met last season and he looked the most likely in the early exchanges.
Castro crashed home a volley after 30 minutes but his celebrations were short-lived as the assistant referee raised his flag for offside.
Betis looked more threatening in the second half and introduced Arsenal striker Joel Campbell in the closing stages, hoping to seize the initiative.
Osasuna rallied late on but despite some pacy wing-play from Roland Lamah they failed to create any clear-cut opportunities.
A disappointing stalemate keeps the home side rooted to the foot of La Liga while Betis stay in fifth position.