The 50-year-old Weber, from St. Ann, Mo., is trying match the late Earl Anthony's record for major victories with 10 and complete the PBA's Triple Crown an unprecedented second time.
Weber, fourth on the career PBA Tour victory list with 36, averaged 236.71 and won 15 of his 24 matches to finish with a 48-game total of 11,812 pins, topping Belmonte by 46 pins for first place.
"When we're done Sunday, I'll talk about those things," Weber said. "I thought I was pulling away from everyone, but all of a sudden, here comes Belmo. I would have been content with second, but I'm happy I only have to bowl one game Sunday."
Weber credited his older brother, Rich, for identifying a minor tweak in his timing that has made a big difference in the way he has bowled the past two months.
"(Rich) watches me bowl a lot and he never says anything if I'm bowling well," Weber said. "He hasn't said anything to me in a couple of months, so I must be doing things right."
Belmonte, who won his sixth career title and first major title in the U.S. Bowling Congress Masters in February, had a 14-9-1 match-play record and 11,766 pinfall total.
Defending champion Sean Rash of Montgomery, Ill., qualified third with 11,661 pins. Tommy Jones of Simpsonville, S.C., was fourth, 25 pins ahead of Finland's Osku Palermaa. Palermaa beat Bill O'Neill of Langhorne, Pa., in their final match, 198-194, to hold onto the final spots in the finals.