One man died after the conditions brought heavy rain and winds of nearly 100mph to parts of the UK and Ireland.
Hundreds of engineers and tree-cutting specialists are working to reconnect 8,500 households in Scotland, said Scottish and Southern Electricity Networks (SSEN).
But the network operator for the north of Scotland said, as of 8.30am on Monday, its teams had managed to restore power to 80,000 other households.
SSEN said engineers had been working in “challenging” conditions after the storm swept in on Friday.
Most of the homes waiting to be reconnected were in the Highlands, along with some communities in Argyll and Bute.
About 2,500 homes in the Republic of Ireland and 100 in Northern Ireland were also without power overnight.
SSEN said repairs had been made more difficult due to storm debris and fallen trees.
The power firm said staff, along with volunteers from the British Red Cross, are visiting “our most vulnerable customers” as part of a “comprehensive welfare operation”.
“Several food outlets, which served more than 9,000 meals to our customers yesterday, are serving hot food this morning in affected communities,” it added.
Meanwhile, most rail lines have been running in Scotland on Monday – but some in the Highlands are still affected by storm damage.
ScotRail said it expects the majority of services to run, but advised customers to check before travelling.
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Ross Moran, route director at Network Rail Scotland, said: “Storm Amy really left its mark on Scotland’s railway.
“With more than 420 incidents, including fallen trees and debris across the tracks, the scale of the recovery effort has been immense.
“Our teams have worked tirelessly, day and night, to clear routes and carry out essential repairs. Thanks to their dedication, we’ve made real progress but there’s still more to do.”
The Met Office said the “swirl of cloud” associated with Storm Amy has now moved away towards Scandinavia.
