Ireland's earlier four-try, 28-8 win over Scotland had left England needing a bonus-point victory in Paris to keep their title hopes alive.
It is Ireland's third title in five years under Joe Schmidt but Johnny Sexton said celebrations were "muted".
They will win the Grand Slam if they beat England next Saturday.
"It's a strange feeling to win the championship with a game to go," said Irish fly-half Sexton.
"We know how difficult it will be [to win the Grand Slam]. The shoe is on the other foot from last year [when Ireland ended England's Grand Slam hopes with victory in Dublin on the final weekend] and I am sure they will be licking their lips."
If Ireland win in London they will secure the third Grand Slam in their history, following previous triumphs in 1948 and 2009.
"It's going to be a big game for us now," said Ireland captain Rory Best.
"Whenever you put yourself in the position to win everything in the Six Nations it becomes massive, but the key for us it approach it as we do every game.
"We are going to have to save the best till last. That's what it's going to take to win everything next week."