The Category Two storm barrelled in off the Coral Sea, making landfall at around 5:00 pm on Wednesday (0700 GMT), government meteorologists said.
Damaging winds of up to 113 kilometres (70 miles) per hour were recorded as Jasper hit the coast, while meteorologists said pummelling rains could swamp some areas with flash flooding.
Rough surf, strong winds and heavy rain pelted the seaside town of Palm Cove, where usually bustling restaurants and high-end hotels battened down the hatches and waited for the storm to pass.
Uprooted vegetation littered the town's coastal promenade and rows of coconut palms and melaleuca trees bowed under the strain of Jasper's gusts.
"It's over the next few hours that we are expecting to see that heavy rainfall really picking up," government meteorologist Miriam Bradbury said Wednesday afternoon as the storm loomed at sea.
"As the tropical cyclone nears the coast, that's when we are going to see the most dangerous and most impactful weather developing."
The cyclone "slowly" made landfall near the largely Aboriginal settlement of Wujal Wujal, the Bureau of Meteorology said, whipping up "destructive wind gusts".
The tourist cities of Cairns and Port Douglas -- both gateways to the Great Barrier Reef -- were also in the path of the storm.
Deputy Premier Steven Miles had warned that "dangerous and life-threatening" floods could linger for "days to come".
About 15,000 homes were left without electricity as winds picked up Wednesday afternoon, utility companies and the Queensland state government said.
Authorities are warning of damage to fences, roofs and other property -- and had urged residents to tie down loose items before the storm arrived.
Flood watches are in place for rivers across the region and power has preemptively been cut to areas where damage is expected.
Four government weather forecasters had to be plucked from a remote offshore monitoring station as Tropical Cyclone Jasper intensified late last week.
An Australian naval destroyer was dispatched to evacuate the meteorologists from Willis Island, which lies about 450 kilometres east of the Australian mainland.
The forecasters were dropped off in Sydney by the HMAS Brisbane guided-missile destroyer on Tuesday.
"Waiting out Severe Tropical Cyclone Jasper on Willis Island was not something we wanted to take a chance on," forecaster William Tom said on Wednesday.
