Continued Irish pressure on the English line resulted in the only try of the game on 24 minutes when Henderson popped off the back of the maul to score, with Sexton slotting over the resultant conversion.
Ireland failed to turn their continued dominance of the half into points and had to be satisfied going into half-time holding 10-3 lead.
England came more into the encounter in the second period and narrowed the deficit through another Farrell penalty but Ireland restored their seven-point lead on the hour mark through Sexton after the visitors were penalised for a high tackle.
Both sides exchanged further penalties to move the score to 13-9 but Ireland held firm in the final minutes, playing the final stages of the match in the England half to secure a win that seals second place in the Six Nations, a world ranking of fourth and top seeding at the next World Cup.
"Big lessons to learn today," England captain Dylan Hartley said. "We set out to win the tournament and we've done that. Obviously disappointed not to win this final game because we had high hopes, we had high expectations of ourselves.
"It goes to show we're not quite there yet as a team. Full credit to Ireland, (they) put us under unbelievable pressure today and gave us one hell of a Test match.
"We're not the finished article, this will keep us grounded, plenty of work to do. But full credit to Ireland today and this atmosphere was a great one to play in."