Norfolk won the opening set 6-3 but Weinberg fought back to take the second 7-5, as Norfolk's serve faltered, resulting in 12 double faults.
Weinberg took the final set 6-0 to book a place in the semi-finals after two hours and 16 minutes.
He said: 'I expected a very tough match. I'm very, very happy. I couldn't be happier to defeat him in front of his home crowd.'
David Wagner of the USA progressed to the last four with a 6-3 6-0 win over Israel's Boaz Kramer in one hour and six minutes.
Kramer's compatriot Noam Gershony dropped just two games in a straight-sets win over the USA's Bryan Barten, and Nicholas Taylor hit back after losing the first set against Sweden's Anders Hard, the USA player taking the match 4-6 6-4 6-1.
Defending women's Singles champion Esther Vergeer produced another masterclass as she overcame quarter-final opponent Sakhorn Khanthasit of Thailand 6-1 6-0 in 52 minutes.
The Netherlands' Aniek van Koot and Jiske Griffioen followed compatriot Vergeer into the last four with wins over Japan's Yui Kamiji and Lucy Shuker of Great Britain, respectively.
Germany's Sabine Ellerbrock edged out Marjolein Buis in three sets.
Vergeer and Buis then teamed up in the Doubles to defeat Angelica Bernal Villalobos and Johana Martinez Vega 6-0 6-1 to reach the semi-finals.
There were also wins for Van Koot/Griffioen, Shuker/Jordanne Whiley and Khanthasit/Ratana Techamaneewat.
In the men's Singles, top seed Stephane Houdet of France overcame the Netherlands' Tom Egberink 6-2 6-2 in the third round.
Defending champion Shingo Kunieda of Japan is also safely through after defeating Great Britain's Dave Phillipson 6-0 6-2.