ESB networks said those customers who remained without power overnight will have their supply restored by noon today or sooner.
As of Saturday evening, 14,000 customers were without power, but the ESB said that a large cohort of these had supply restored in the last few hours.
Around half of these were affected by large outages in the Swords and Malahide areas of North Dublin, and should have had power restored by nightfall, ESB Networks said in its latest update on Saturday evening.
Crews were to continue to work as long as is safe and would be mobile again at first light.
A spokesperson said that those without power overnight would be in "the low thousands", and would have electricity restored in the morning.
All estimated restore times are up to date on Power Check and generally indicate noon tomorrow for those who will not have power restored tonight.
Earlier the spokesperson for ESB Networks said that north Dublin, Wicklow and Wexford would be the areas most affected by overnight outages.
Storm Betty caused flooding and made roads dangerous with fallen trees, branches and debris, as it crossed much of the south of the country last night and into the early hours of this morning.
Storm force winds caused damage to the power network overnight, at one point affecting more than 70,000 homes, farms and businesses.
In a statement this evening, ESB Networks said its crews have been "working since first light to restore electricity to homes, farms and businesses following storm force winds and lightning associated with Storm Betty which caused damage to the network".
"All available resources have been deployed and ESB Networks crews will continue to work late into the night to restore electricity," the statement added.
The company apologised to those impacted for the disruption to family and commercial life the outages caused.
"Customers without power can check for real-time updates on when their fault is expected to be repaired at www.PowerCheck.ie."
Workers and volunteers at the famous Clontarf Public Baths in Dublin said that their local "paradise" has been left "destroyed" by the impact of Storm Betty.
A 94-year-old woman escaped injury when a large tree in her front garden fell during the storm, narrowly missing her home in Rathmines, Dublin city.
While a taxi driver escaped injury in Templeogue in south Dublin when a tree hit his car.
A number of trees fell on the rail line between Drogheda and Dundalk, with more trees down north of the border, and the 7.35am Dublin to Belfast train terminated at Portadown.
All Met Éireann weather alerts have now been lifted.
In Cork city, the council's roads, drainage and fire crews were busy, with Cork City Fire Service tackling seven calls yesterday evening and 15 overnight.
These included three crashes, including a car colliding with a truck on the Airport Road. No-one was seriously injured in the incident.
The Glasheen River burst its banks in the Riverview Estate in Glasheen, threatening a number of houses and there are reports of a least one house flooded on St Mary's Avenue on the northside.
There was also a river rescue at Anderson Quay.
Cork City Council's David Joyce said that there were a number of significant events across the city, but that the impact of Storm Betty was not widespread.
Much of the flooding in the city had abated by 10pm.
They were on stand-by today to respond to any overnight reports and clean-up calls.
In Co Cork, the clean-up is under way in Carrigaline, following serious flooding along the town's main street.
While in the west of the county, around Castletownbere and Allihies, there were widespread reports of debris including trees on the roads.
In Co Kerry, the N70 road from Waterville to Cahersiveen was closed due to flooding, and maintenance crews removed trees blocking the N86 at Camp, with a number of Féile Thrá Lí Festival events cancelled or moved indoors yesterday.
Sailors rescued
This morning the Castletownbere lifeboat was launched in gale force winds to go to the assistance two sailors whose yacht got into difficulties on the northern shore of Kenmare Bay in Co Kerry.
The 31-foot ketch had been anchored off Dromquinna and, as the storm became increasingly strong, the vessel dragged anchor and became stuck on a rock.
The sailors were concerned and raised the alarm with the Irish Coast Guard's Marine Research Co-ordination Centre in Valentia stating that they were requesting immediate assistance.
The yacht was subsequently located on a rock south of Dromquinna but, with the rising tide, volunteer lifeboat crew were able to attach a tow rope and pull the vessel clear.
Additional reporting: Jennie O'Sullivan, Jason Kennedy, PA
