Ireland dispatched Scotland, but it was France's victory over England in Paris that secured the championship for the Irish.
A revitalised France beat an uninspired England 22-16 on Saturday to claim a rare victory over their arch-rivals whose defeat handed the Six Nations title to Ireland.
Les Bleus, who had lost nine of their last 11 meetings against England in the Championship, prevailed through penalties by Maxime Machenaud (4) and Lionel Beauxis and a penalty try while their opponents scored two penalties and conversion by Owen Farrell, a penalty by Elliot Daly and a try by Jonny May.
Ireland top the table with 19 points from four games after securing a 28-8 bonus-point victory over Scotland earlier on Saturday, meaning they cannot be caught by England when the two teams meet at Twickenham next weekend.
France, who narrowly lost their first two matches against Ireland and Scotland, defended furiously throughout and it became obvious that England, who needed a bonus-point win to keep their title hopes alive, would not score the four tries they needed at Stade de France in Paris.
It was England's second defeat in a row after they lost to Scotland in the previous round of matches.
The championship is Ireland's third in five years.
But the bigger prize would be beating England next Saturday for Ireland to earn only a third Grand Slam, beside those in 1948 and 2009.
Scotland upended England last time out but that was at Murrayfield. With an awful record away from home, the Scots' own championship bid was almost inevitably squashed by an Irish side which owns the visitors in Dublin, where Scotland has won only once in 20 years. Scotland had its chances on Saturday but wasn't as accurate, and butchered two tries, probably three.
By the time new cap Blair Kinghorn claimed Scotland's sole try, Ireland had three, two to wing Jacob Stockdale.
The bonus-point fourth try came with 11 minutes to go, when hooker Sean Cronin, only three minutes after replacing Best, dived over from the back of a rolling maul.
"We are three or four years behind Ireland in terms of what they've done and achieved over the last few years," Scotland coach Gregor Townsend said.
"We need to make sure that when we have these experiences that are tough, it goes into our work to improve. We play Ireland next year in the Six Nations and we will play them in 18 months at the World Cup and we need to make sure we are a better team when we play them."
France, who narrowly lost their first two matches against Ireland and Scotland, defended furiously throughout and it became obvious that England, who needed a bonus-point win to keep their title hopes alive, would not score the four tries they needed.
It was England's second defeat in a row after they lost to Scotland in the previous round of matches and only their third in 27 matches under coach Eddie Jones.
France coach Jacques Brunel praised his team's defensive performance.
"Our defence was remarkable," Brunel told a news conference after his fourth game in charge.
"We knew that the physical challenge would be key. We defended well and we always went forward."
France were superior at the breakdown, an area where England also suffered when they lost to Scotland in their previous game.
"We needed to be good at the breakdown, and maybe they were not used to be bothered in that area," said Brunel.
"If they are free in that zone, it becomes complicated for their opponents."
"The way we defended, I had never seen that," said wing Remy Grosso, who was named man of the match.
"It's a very big moment, it's a big emotion to beat England."
France, who had lost nine of their last 11 Six Nations games against England, are third in the standings behind England on points difference.
Wales face Italy later on Sunday, but the outcome of that match will have no impact on Irish fortunes.