Two early Owen Farrell penalties put England ahead before Ireland hit back with a well-worked Dan Sheehan try.
England's hopes of an upset were dashed when Freddie Steward was sent off just before half-time.
Robbie Henshaw and Rob Herring tries either side of Sheehan's second of the game ensured Ireland's first Grand Slam in five years, and first won in Dublin.
England, much improved from their humiliating defeat by France at Twickenham last week, scored a deserved second-half try through Jamie George, but were unable to pull off a major upset and ruin Ireland's big day.
Despite being tested by a resilient English side, Ireland - as they have so often done under Andy Farrell - found a way to win as they gave talismanic captain Johnny Sexton the perfect send-off in his final Six Nations match, even though his day ended early because of injury.
The full-time whistle was greeted with Irish celebrations at a jubilant Aviva Stadium as the home side delivered on their immense promise, having entered the competition as the world's number one side.
With the prospect of being confirmed as champions before kick-off ended by France's 41-28 win over Wales, the tension inside the Aviva Stadium was palpable from the start as Ireland plotted the dream finale to a memorable campaign.
But while few gave England much chance of spoiling the Irish party following their record-breaking hammering by France last week, it was the visitors who settled quicker.
Their reward was two penalties, both scored by Farrell, the second coming after Alex Dombrandt stopped Johnny Sexton from scoring with a crucial tackle at the other end.
Clearly nervous, Ireland were made to wait until the 18th minute for their first score. But it was a big one as Sexton thumped a penalty between the sticks to surpass Ronan O'Gara as the competition's record scorer.
While Sexton's record-breaking penalty settled a few nerves, Sheehan's score produced a massive roar from the Aviva crowd, the hooker crossing after a well-worked set-piece that saw Josh van der Flier breaking from a line-out maul.
With Sheehan having given Ireland a much-needed shot in the arm, things went from bad to worse for England on the cusp of half-time when Steward was sent off after colliding with Hugo Keenan.
After an Irish pass went forwards, Keenan stooped to pick up the loose ball. The on-rushing Steward turned sideways at the last moment and Keenan's head collided with the England full-back's arm, with referee Jaco Peyper ruling that the contact merited a red card.
Despite their numerical disadvantage, England refused to lie down in the second half, and moved to within a point when Farrell booted his third penalty between the posts.
But Ireland managed to regain control and exerted enough pressure to prise open holes in the English defence, with Henshaw able to slip through before Sheehan's second score seemingly put the hosts out of England's reach.
To their credit, England continued to plug away and scored their only try when George crashed over, but after the visitors' Jack Willis was sin-binned, replacement Irish hooker Herring stretched to score his first Six Nations try and complete a famous triumph for Ireland.
While England head coach Steve Borthwick can take heart that his side delivered on their promise to show more fight than they did against France, a fourth-place finish is a stark reminder of the task facing Eddie Jones' successor.
In contrast, Ireland will now approach the World Cup later this year with relish as they look to progress past the quarter-finals of rugby's quadrennial showpiece for the first time.
That is all to come. For now, Ireland can celebrate retaking their place at the top of the Six Nations.
* Ireland have now won four Grand Slams with triumphs coming in 1948, 2009, 2018 and 2023
* They have won five Six Nations titles with their other wins in 2009, 2014, 2015 and 2018
* Andy Farrell's side set a new national record of eight straight Six Nations wins, surpassing the seven-game streak they set between 2004 and 2005
* Ireland are on a national record run of 14 wins at home
Former England captain Matt Dawson on BBC Radio 5 Live: "I'm proud of the way England fronted up but they gave away far too many penalties.
"Their ill-discipline kept giving easy ball to Ireland. A good performance by England but the focus should be on this tremendous Irish side.
"I just cannot see anyone beating the Irish. They are head and shoulders ahead.
"This Ireland team have what it takes, not just to win the Six Nations, but the World Cup as well."
Former Ireland number eight Jamie Heaslip on Radio 5 Live: "It's a great day for the Irish. They are a special group with a different mindset to the past. They are comfortable with being the number one side in the world."
* 4 February: Wales 10-34 Ireland
* 11 February: Ireland 32-19 France
* 25 February: Italy 20-34 Ireland
* 12 March: Scotland 7-22 Ireland
* 18 March: Ireland 29-16 England
Ireland: Keenan; Hansen, Henshaw, Aki, Lowe; Sexton (capt), Gibson-Park; Porter, Sheehan, Furlong; Baird, Ryan; O'Mahony, van der Flier, Doris.
Replacements: Herring, Healy, O'Toole, Treadwell, Conan, Murray, R Byrne, O'Brien.
England: Steward; Watson, Slade, Tuilagi, Arundell; Farrell (c), Van Poortvliet; Genge, George, Sinckler, Itoje, Ribbans, Ludlam, Willis, Dombrandt.
Replacements: Walker, M Vunipola, Cole, Isiekwe, B Curry, Mitchell, Smith, Marchant.
Referee: Jaco Peyper (SA)
Touch judges: Ben O'Keeffe (NZ) and Pierre Brousset (FRA)
TMO: Marius Jonker (SA)