The announcement by Finance Secretary John Swinney was made as the clean-up operation began in earnest.
He has activated the Bellwin Scheme after representations from Moray Council.
The minister said: “We know, and I have seen for myself, just how devastating the effects of flooding can be for our communities.
“That is why I have today agreed to activate the Bellwin Scheme to provide support to affected councils to assist with immediate and unforeseen costs of dealing with the aftermath of the severe weather.
“As the full impact on local communities continues to be assessed, we stand ready to provide urgent assistance and the Scottish government will consider applications from local authorities for emergency funding through the Bellwin Scheme.”
Elgin’s £86m flood alleviation scheme, which is currently being constructed, was put to a severe test as the north of Scotland was hit by the remnants of the former hurricane on Sunday and Monday.
Two hundred homes were evacuated at one stage on Monday over fears the River Lossie would burst its banks where the project has not yet been completed.
Most returned home as the water level went down.
Moray Council leader Allan Wright said: “It is calming down. Two or three families had to be put in overnight accommodation, particularly families with young children.
“The rest were allowed home at about 9.30pm.
“The flood defences had held. It was a close run thing because the flood scheme in Elgin has not yet been completed.
“But it held and we all breathed a sigh of relief before we went to bed.”
Properties in parts of Aberdeenshire were also evacuated as river water levels rose.
In Huntly, residents in local care homes were moved to the town’s Jubilee Hospital, while about 150 people staying in a caravan park in Ballater were also evacuated.
The storms wreaked havoc across the region, closing roads and rail links.
Nick King from Network Rail said: “Trains were running from Inverness to Aviemore on the Highland line. But at the moment they are not going further north than Perth.
“On the Aberdeen-Inverness route, issues are still around the Elgin area where we still have fairly significant flooding and several structures around the River Lossie which need to inspected by divers before they can be safely opened.
“Trains are running from Inverness as far as Nairn. And from Aberdeen as far as Huntly. But at the moment we are expecting it to be tomorrow before we are able to reopen either of those routes in full.”
He added: “The situation is much improved on what it was yesterday where we had up to 3ft of standing water on the railway, on the Highland line and on the Aberdeen-Inverness route.
“Most of that water has now cleared and what engineers are now doing is assessing the damage that has been done.
“Around Kingussie on the Highland line we have had sections of ballast, the track bed essentially, washed away. That’ll need to be repaired. We have also damage to structures by the sheer force and volume of water hitting them.”
“Essentially we are working towards reopening the railway tomorrow. At the moment that is a target date because there are quite significant infrastructure faults that we need to fix.
“It is principally about cleaning off the tonnes of mud and silt and rocks and other debris that has built up on the lines.
“Also, as I said, dealing with a wash-out of the track bed at Kingussie.”
Flooding expert Richard Brown of Sepa said: “There will be a lot of residual impacts from yesterday right across Morayshire/Aberdeenshire and across into the Highlands and, indeed, there has been quite a bit of rain overnight in the west Highlands, about another 30mm or so.
“But that really pales into insignificance compared to some of the totals we saw from Sunday afternoon through until early Monday morning which caused the fairly serious flooding events that we saw yesterday.”
He told Good Morning Scotland: “The situation overnight has very much improved. Most of the rivers are dropping back from the peaks of yesterday afternoon.
“Although one or two are still rising, the Don for example in Aberdeenshire. But there are no particular cause for concern at the present time.”
Richard Brown, head of hydrology at the Scottish Environment Protection Agency (Sepa) said there were still 38 flood warnings in force across a “wide swathe of northern Scotland” and some down into Tayside.
He added: “However that is a much improved situation from yesterday. We had well over 40 at the peak of the events yesterday afternoon.”
The A835 south of Ullapool is partially open with a convoy system. This is the main road to Inverness and will affect traffic for the Stornoway ferry.
Quite a few smaller roads in Elgin are still affected by flooding but the situation is calming down weather wise, water is receding.
BBC weather forecaster Judith Ralston said there would be rain at first for many today, clearing southwards and turning showery. There will be blustery showers to follow, especially in the north, but brightening up later.
Showers will become more frequent over North West. Highs of 18C for the northeast and 16C for most, but it will feel cool with a fresh to strong westerly.