Joe Schmidt's Irish team secured a top seeding for the 2019 World Cup draw as a try by lock Iain Henderson after 23 minutes proved the difference between the sides.
While England improved after being completely dominated in the first half, they failed to make a serious dent in the Irish defence.
England won the tournament last week with a 61-21 thrashing of Scotland, but they collected the Six Nations trophy in a downbeat mood as the third English side in 16 years to be denied a grand slam in Dublin on the final day of the tournament.
"We didn't slam it but, hey-ho, we're still the champions," said England captain Dylan Hartley. "Full credit to Ireland. They put us under unbelievable pressure."
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England lost in 2001 and 2011 but hammered Ireland in a grand slam decider in the old Lansdowne Road in 2003, months before they won the World Cup.
The win also denied England a tier-one record of 19 consecutive wins just four months after Ireland prevented world champions New Zealand from reaching the same milestone with a 40-29 win in November.
England, who matched the All Blacks' 18-match run with their victory over Scotland last week, have now lost seven times in their last 10 visits to Dublin.
Victory rescued a poor Six Nations championship for Ireland, who lost to Scotland and Wales after beating three southern hemisphere teams last year.
Ireland ended the tournament in second place and are guaranteed to be one of the top four seeds in the draw for the 2019 Rugby World Cup, which will see them avoid heavyweights New Zealand, Australia and England in the group stage.
"It wasn’t about us stopping England. It was about us producing a performance. This is a really big lift for us,” Ireland captain Rory Best said.
"England are a fantastic team, they deserve the Championship."
(Editing by Ed Osmond)
