Ireland have 19 points from five matches having completed their campaign with four victories, and must now hope Wales upset England in Cardiff and Scotland beat France in Paris later on Saturday to claim a third Six Nations crown in a row.
Fullback Hugo Keenan also crossed for a try as Ireland failed to finish numerous other entries into the Italy 22 in a scrappy display that was a last in international rugby for retiring loose-forward Peter O'Mahony and scrum-half Conor Murray.
Italy played for half an hour with 14 players after yellow cards for Michele Lamaro and Giacomo Nicotera, and a 20-minute red for Ross Vintcent, but managed tries through Australian winger Monty Ioane and scrum-half Stephen Varney in a spirited end to their campaign.
"Relief is the overriding feeling," Ireland captain Caelan Doris admitted afterwards.Â
"It was a proper battle that went down to the wire. We knew it would be tough, we were ready for it. Right to the last five minutes they tested us.
"You talk about ugly wins and finding a way when things are not going your way and we did that. We got the five points and the rest is out of our hands now."
Italy scored first as flyhalf Paolo Garbisi's perfect grubber kick sat up for Ioane to gather and cross in the corner, but Ireland responded with a simple backline move from a five-metre scrum that sent Keenan over the tryline.
Tommaso Allan's long-range penalty gave Italy the advantage again, but Ireland led 12-10 at halftime as, after Lamaro's deliberate knock-down at a ruck and subsequent yellow card, the visitors mauled their way over for Sheehan to score.
Ireland camped in the Italy 22 at the start of the second period and were rewarded with a second score - a carbon copy of their first - as Sheehan was at the back of a rolling maul.
Vintcent received a yellow card, later upgraded to a 20-minute red on bunker review, for head-on-head contact with Keenan as the replacement flanker was too upright in the tackle.
Sheehan scored his third try when he collected Australian Mack Hansen's tap-back from a Jamison Gibson-Park kick to complete his hat-trick.
Ireland should have been out of sight, but Jack Crowley's three missed conversions kept their hosts in the game.
And when Varney scored Italy's second converted try following a brilliant break from Ange Capuozzo, the gap on the scoreboard was only five points with 18 minutes remaining but Ireland hung on for the win.
