Ireland will retain their title unless England can beat France by 26 points at Twickenham in the tournament's final match later on Saturday.
"We knew what we had to do at halftime and we executed it very well," Ireland flanker Sean O'Brien told the BBC.
"It was important we expressed ourselves. We had to win the game first and foremost and we stuck to our guns and got the result.
"We're going to wait very anxiously. Our eyes will be glued to the England game."
Forwards Paul O'Connell and O'Brien scored early tries to put Ireland 17-3 ahead but Scotland flyhalf Finn Russell scampered over in the left corner to reduce the deficit to 20-10 at halftime.
Ireland dominated possession and were rewarded when centre Jared Payne scythed through the Scottish defence to touch down and Scotland were on the ropes when prop Geoff Cross was sent to the sin-bin.
But Jonny Sexton missed two straightforward penalties before re-adjusting his radar to extend the Irish lead to 33-10 and O'Brien burrowed over from close range seven minutes from time.
Scotland fullback Stuart Hogg failed to ground the ball over the line under heavy pressure from Jamie Heaslip, leaving Ireland to celebrate their record victory at Murrayfield.
Earlier in Rome, Wales spluttered through the first half and led only 14-13 at the interval against a determined Italian side.
But the Welsh produced a scintillating display of attacking rugby as winger George North crossed the line three times and Liam Williams, Rhys Webb, Sam Warburton and Scott Williams also went over to put Wales ahead 61-13 in the dying minutes.
There was a late sting in the tail for the inspired Welsh, however.
Italy winger Leonardo Sarto burst down the left, evaded the tackle of Dan Biggar and touched down in the corner before Luciano Orquera kicked the touchline conversion to give Ireland and England a late lifeline.
Wales centre Jamie Roberts scored the game's opening try and Giovanbattista Venditti went over for Italy who started brightly and were scenting a famous victory at halftime before being swamped by a red tide.
England host France at Twickenham in London at 1700 GMT.
(Editing by Toby Davis)
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