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Sexton added another penalty as Ireland were winning the battle for territory, before England responded with a drop goal from George Ford.
The game was evenly poised with 10 minutes to play in the first half, but Stuart Lancaster's side started to commit costly errors which allowed Ireland penalties in kicking distance.
Sexton added his third kick to give his side a six-point lead, but missed from close range to spurn an opportunity to take a nine-point advantage into the break.
However, Ireland upped the ante in the second half and won yet another penalty which was converted by the boot of Sexton as Schmidt's men took control of the game.
Matters got even better for Ireland when Connor Murray lifted a kick over the England line towards the corner, and Robbie Henshaw leaped over Alex Goode and touched to ball down to score the game's only try.
Ford kicked a penalty to bring England within 13 points, and then minutes later added another from close range to narrow the gap.
Lancaster's side began to find their rhythm in the dying minutes of the game, but they were unable to power over the line from yards out as Ireland held out to secure the crucial win.