Neil Back's men raced into a 24-3 lead with England flanker Hendre Fourie, Rhys Oakley and Peter Wackett all touching down.
The Saints, who opted to rest several of their big guns including England lock Courtney Lawes, hit back through Phil Dowson before they were awarded a penalty try.
Ashton's converted score brought the Heineken Cup finalists to within two points before three Stephen Myler penalties saw them home.
Newcastle, then, survive despite going down to a 42-12 defeat in which former England lock Danny Grewcock played his last game before retirement.
At the other end of the table, Leicester came from behind to beat London Irish 32-23 and secure top spot.
England hopeful Topsy Ojo had latched onto the two intercept tries for the Exiles but the Tigers came roaring back with England flanker Tom Croft among their five tryscorers.
They will face the Saints in the play-offs while Saracens, who secured their 11th straight win in a tight 16-13 defeat of London rivals Harlequins, will host Gloucester, who finished their regular season with a stunning 68-17 defeat of Sale.
Elsewhere Exeter leapfrogged Wasps into eighth place following their 21-8 victory over the crisis club.