Needing a bonus-point victory to apply pressure on Championship favourites France and fellow contenders England, Ireland overcame a slow start to score four tries and secure a 16th straight win against the Azzurri.
It pushed them to the top of the Guinness Six Nations table on 19 points but England later usurped them with a thumping win against Wales.
Ireland were far from their brilliant best in Rome but Sheehan proved the difference with two tries from rolling mauls and another from an excellent Mack Hansen assist.
Italy, who played with 14 men for almost half the match thanks to one red and one yellow card, scored the first try through Monty Ioane before Hugo Keenan responded and Sheehan scored three.
Stephen Varney replied for Italy and the deficit was just five points as the game headed into the final 10 minutes but Ireland held on for a crucial win.
The Stadio Olimpico was a see of green at kick-off, with Ireland fans travelling in their droves to see what they hoped would a Grand Slam coronation when the fixtures were announced.
Last weekend’s defeat to France dashed those dreams but there was still an outside chance of winning the title, and the final international matches of legends Conor Murray and Peter O’Mahony.
Italy may have shipped 120 points in their last two matches but they have also scored some incredible tries in this Championship, and they added another to what will be an epic 2025 montage in the 12th minute after a scratchy start to the game.
Tommaso Menoncello broke through a tackle mid-way inside Ireland’s half and off-loaded to Paolo Garbisi, springing the fly-half into space. Garbisi, surrounded by green jerseys, then showed brilliant poise while being chased down and prodded through a delicious kick for winger Ioane to chase and dot down. Tommaso Allan’s conversion from tight to the left touchline made it 7-0.
Inside centre Menoncello wreaked havoc in the first quarter, with a line break from his own half carrying the Azzurri 50 metres upfield and into Ireland’s 22 before a knock-on ruined an excellent try-scoring chance.
Ireland’s start was sloppy but their forwards helped to restore parity mid-way through the half when they lay siege to the Italy defence five metres out and, from a scrum, Keenan sprinted up from full-back to catch a short pass and blast his way over for a try.
Injuries piled up for Italy in the first half, with second-row Dino Lamb and back-rowers Seb Negri and Lorenzo Cannone forced off inside 30 minutes. However, they were next to score as Allan booted a brilliant penalty from long-range marginally over the cross-bar after Garry Ringrose was caught offside.
Ireland were still guilty of making uncharacteristic errors but they grabbed the half-time lead by capitalising on an unforced Italy mistake. Replacement flanker Michele Lamaro was sin-binned for a deliberate knock-on when he threw out his hand to block Jamison Gibson-Park from a pass, and Ireland made him pay with a second try from the penalty.
A driving maul shoved the Azzurri back over their own line and Sheehan dotted it down. Jack Crowley missed the conversion but the visitors led 12-10 at the break, and they headed for the dressing room relieved to be ahead but knowing there was much work to do.
Two tries was the minimum target for the second half and they scored the first of those within eight minutes of the re-start. After a couple of near-misses near the Italy line, Ireland eventually got the ball down as the driving maul worked a treat again. Jack Conan won a five-metre line-out and Ireland drove over the line, with Sheehan scoring his second.
It got worse for Italy soon after as replacement flanker Ross Vintcent was sin-binned for a high tackle from the re-start, reducing the hosts to 13 men for seven seconds before Lamaro was re-introduced. Vintcent’s yellow was later upgraded to a red after a bunker review and he was off for 20 minutes.
The biggest cheer for the day was reserved for the introduction of replacement Ireland flanker O’Mahony in his last of 114 caps, while scrum-half Murray joined him for cap No.126 in the second half as Ireland piled on the pressure.
After a sticky start, the momentum was with them and the all-important fourth try followed soon after as Hansen batted a Jamison Gibson-Park cross-field kick down to a supporting Sheehan to score.
Crowley’s kicking problems continued, as he missed a third conversion from four attempts, to leave Ireland’s lead at just 12 into the final quarter. And that was quickly cut to just five as Varney scored a second Italy try following a neat Ange Capuozzo chip in behind Ireland’s defence.
Vintcent returned for the final 10 minutes and Italy threatened to win it when Leonardo Marin made a line-break and carried into the Ireland 22. But Garbisi lost the ball, and Ireland saw it out for a narrow win.