The storm came ashore about 25 kilometers (15 miles) northwest of the resort city of Cabo San Lucas on the Baja California coast, gradually weakening as it approached land.
Norma was moving northeast at 9 kph, packing winds of up to 110 kph, the National Hurricane Center (NHC) reported at 0000 GMT.
Norma was expected to bring rainfall of up to 45 centimeters (18 inches), producing "flash and urban flooding, along with mudslides in areas of higher terrain," the NHC said.
"Heavy rains and flash flooding to continue through the weekend," it said.
"Life-threatening surf and rip current conditions" would affect the coast of southwestern and west-central Mexico and Baja California Sur for the next couple of days.
Fueled by warm Pacific waters, Norma had intensified to a Category 3 hurricane on the Saffir-Simpson scale, and was still a Category 1 when making landfall, before weakening.
Hotel employees in Cabo San Lucas urged guests to stay indoors until authorities give the all-clear, Gustavo Matamoros, a hotel worker, told AFP.
Local authorities say about 60,000 tourists are staying in the area, most of them international visitors.
Footage from Milenio TV showed flooded streets as Norma arrived. No casualties have been reported by authorities.
Norma should continue to weaken as it moves across the Baja California Peninsula toward the west coast of Mexico on Sunday.
Two people died last week when Hurricane Lidia struck the western states of Jalisco and Nayarit.
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