The Waikato side ran in five tries in Hamilton, leading 27-10 halftime, before pulling away late in the second half.
Weber and number eight Pita Gus Sowakula crossed in the final quarter to snuff out any chance of a comeback from the visitors.
Chiefs lock Brodie Retallick was happy with his side's outstanding peformance.
"That was just a great Chiefs' pack performance," Retallick said.
"We knew the Waratahs led the statistics around the ruck and the breakdown so we knew we had to be tough there."
"It's playoff footy and physicality. The team that is the most physical usually comes out on top."
The Waratahs, who lost all 13 games last season, had crossed the Tasman from Sydney determined to continue their resurgence under new coach Darren Coleman.
They scored tries early in each half but handling errors cost them.