Livigno's Snow Park and Aerials & Moguls Park will be key venues during the February 6-22 Games, with 26 gold medals up for grabs in events staged there.
Eliasch sounded the alarm last week about delays and that the organisers were "struggling to make ends meet" because the Italian government had not released any funds.
The venues in Livigno must produce large quantities of artificial snow to create the halfpipes and jumps needed for snowboarding and freestyle skiing events, with their snow cannons being supplied by water from the Monte Sponda reservoir.
Visible progress
"Over the past two weeks, the Local Organising Committee has taken important steps forward. Progress is visible, and the team's determination to overcome challenges and their efforts to deliver should be commended," Eliasch said.
"We want to recognise the dedication of everyone involved - from the Organising Committee on the ground to the national and regional authorities and all partners.
"Their teamwork and perseverance are essential to ensuring a successful implementation of the events in Livigno."
With the Games due to start in February, Eliasch said Italy's challenges were symptomatic of deeper structural issues facing winter sport, as rising costs, climate pressure and under-used infrastructure fuel calls for a rotating model of permanent Olympic hosts.
Livigno's mayor Remo Galli had said the delays were caused by a technical problem and that they would have the snow needed to host the Olympics.
"Our goal remains unchanged: to give our athletes the stage they deserve - a stage worthy of their talent and of the Olympic spirit they embody. A place where they can perform at their best and inspire the world," Eliasch added.
"Guided by this shared commitment, the path toward the Games that will showcase the power of snow sport is taking clearer shape every day."