They were heading for a goalless draw when right-back Dani Carvajal volleyed Karim Benzema's cross into the net.
Benzema wasted a good chance when he fired wide with the game goalless, while Carvajal had an attempt tipped over the bar before scoring.
Betis went close through Nabil Fekir's free-kick.
Real Madrid bid £137m for Paris St-Germain forward
Real are top of La Liga after three games, level on seven points with Sevilla, Valencia and Mallorca.
They have lost 300m euros in revenue because of Covid-19 and have only signed defender David Alaba on a free transfer from Bayern Munich this summer.
However, Real are keen to strengthen their attacking options by signing France World Cup winner Mbappe, 22, before next week's transfer deadline.
PSG sporting director Leonardo said the club would "not hold Mbappe back" but considered Real's £137m bid to be "not sufficient".