The combative striker needed 40 seconds to open the scoring, registering the fastest league goal so far this season.
He capitalized on the keeper's failure to handle Ahmed Abdul-Aziz's cross to head into an empty net at the far post.
Olympic defender Ateya Al-Belkasi, who will join Ahli next summer, leveled the score in the 34th minute with a close-range header after a goal-mouth scramble.
Khaled Yosri restored Contractor's advantage on the hour mark after converting a penalty.
Al-Weeshi wrapped up the victory five minutes later with a brilliant overhead kick after initially miscuing his header.
The 23-year-old, who made a fine impact since joining from Ahli last summer, took his tally to seven league goals.
The victory eased the crisis of Contractors, who had been without a win since beating Telecom Egypt in November.
They moved into eighth place with 21 points, three above the relegation zone.
Olympic's hopes of retaining their top-flight status after sealing promotion last season faded as they lie at the bottom on 13 points.