Weber's second try came after a breath-taking buildup, with replacement back Alex Nankivell bursting out of defence and fullback Solomon Alaimalo punching a hole through midfield before giving off to the pint-sized number nine in the 70th minute.
The Chiefs, who also had tries from flanker Mitch Karpik and Anton Lienert-Brown, held on in a nail-biting finish to keep their slender finals hopes alive.
The Chiefs are in 11th place, three points outside the top eight playoff places.