Seven days after they rallied from 16 points down to earn a 31-31 draw against the Highlanders, the Chiefs again came from behind and this time they snapped a three-game winless streak as Anton Lienert-Brown and Weber went over in the final 12 minutes.
The Chiefs led at the break, Weber and Mitch Karpik scoring tries, though Curwin Bosch's two penalties and Hyron Andrews' try, coming after Sibusiso Nkosi's offload from a high kick, meant the Sharks were only 17-13 behind.
Ruan Botha then crossed to give the Sharks their first lead of the match and they were still in front with 10 minutes to go as Marty McKenzie was unable to add the extras to Lienert-Brown's score, which had been set up by Weber's incisive pass.
But the Chiefs would claim the victory as Weber again provided support to collect Solomon Alaimalo's pass and touch down for the match-winning score.
The losing bonus point was enough to move the Sharks to the top of the South African Conference, with the Jaguares one point behind having also picked up a losing bonus point in going down 32-27 at the Highlanders earlier in the day.
The damage was done in the first half as James Lentjes, Matt Faddes, Sio Tomkinson and Liam Coltman all went over to give the Highlanders a 26-13 advantage.
Marcos Kremer and Ramiro Moyano added to the Jaguares' sole first-half score through Matias Moroni, but the Highlanders held on.