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.
