They led 24-3 after 18 minutes after tries from Hendre Fourie, Rhys Oakley and Pete Wackett.
But the Saints were ahead by 54 minutes as their experienced England players turned the screw.
Leeds earned a losing bonus but finish level on points with Newcastle, but with a poorer points difference.
Their relegation will not be confirmed until May 18 and the outcome of the Championship play-off final.
If Worcester win they will be promoted and Leeds relegated.
But if Cornish Pirates win the Yorkshire club will be reprieved and the Premiership status quo will remain.