Motorists who drive at more than 30km/h over the speed limit will face court prosecution and up to €2,000 fine, under amended proposals brought to Cabinet by Minister for Transport Shane Ross.
Mr Ross told ministers that excessive speeding is a major contributing factor in road deaths
He said it was a lethal problem that could best be tackled by introducing a new principle: "The faster you go - the more you get punished."
Under the existing law, a motorist found to have broken the limit is fined €80 and hit with three penalty points, irrespective of their speed.
Under Mr Ross's proposals, a gradually increasing scale of penalties would see a driver found to be less than 10km/h over the limit facing a €60 fine and two penalty points.
Anyone speeding between 10km/h and 20km/h over the limit would have to pay an €80 fine and collect the penalty points.
Anyone speeding between 20km/h and 30km/h over the limit will get a €100 fine and four penalty points.
The most severe sanction would be reserved for a person driving in excess of 30km/h above the limit, they would face a court prosecution and up to a €2,000 fine.
Repeat offenders would face jail.
Speaking after the Cabinet meeting, Mr Ross said there was "an issue of enforcability" of his plan for graduated speeding points.
On RTÉ's News at One, he said there "are difficulties with people not handing in their licences, and there are problems with the court service".
He said enforceability was the next step after the legislation was backed by Cabinet.
The minister said there was good discussion at the meeting and that it "wasn't heated and was very constructive".
Earlier today, Mr Ross said it was a "balanced and moderate piece of legislation" that would see minor offenders get more lenient points and smaller fines, while major offenders who drive at really high speeds would get harsher punishments.
He said it was about changing attitudes and he wanted to people to stop driving at dangerous speeds.
His original plan had been redrafted by a Cabinet subcommittee, and Mr Ross has said he accepted it was worth changing to ensure more lenient penalties were imposed on motorists found to have been only marginally over the limit.
The Fine Gael chair of the Oireachtas Transport Committee said he "stands on the side of saving lives" and said graduated speed fines were sensible.
Speaking on RTÉ's Today with Sean O'Rourke, Fergus O'Dowd said the reality was that road deaths were increasing and that 131 people have died this year compared to 124 this time last year.
Of the 93 people who were killed in August of this year, he said, 70 had died on rural roads where there is a maximum speed limit of 80km/h.
Fianna Fáil's transport spokesperson Marc MacSharry said his party was not opposed to anything that saves lives on roads but the current proposals would not work.
Speaking on the same programme, Mr MacSharry said that more time should be given to reviewing speed limits around the country and studying what has worked in other jurisdictions.
In addition he said, some consideration should be given to adequate segregation for cyclists.
He added that there was a backlog of 90,000 court cases, as well as €27m outstanding in fines, as a result of the "poorly thought-out fines act passed in 2014".
Meanwhile, the Department of Transport is to begin a process of reviewing speed limits around the country.
The development will involve consultation with engineers in local authorities to analyse existing limits.