Carlos Tevez had initially looked to once again be the scourge of his former Manchester City coach Mancini when putting Juve ahead in the fifth minute, but Icardi secured Inter a share of the spoils just after the hour mark.
Mancini famously fell out with Tevez on the touchline during a 2011-12 UEFA Champions League match against Bayern Munich, and the Argentinian looked particularly delighted to prod Juve into the lead early on.
Inter, who included new signing Lukas Podolski on the bench, saw plenty of the ball in the first half, but the hosts looked much the more likely before the interval, as Arturo Vidal and Paul Pogba both went close.
Juve ended 2014 disappointingly as they won just one of their final five games in all competitions and, although they continued to look bright after the break, Icardi levelled with a clever finish in the 64th minute.
Both sides piled men forward in an extremely open final 15 minutes, with Icardi spurning a glorious opportunity with nine minutes to go.
And while Mateo Kovacic earned a late red card, Juve were unable to capitalise and their lead at the top of Serie A was cut to just one point after Roma's controversial win over Udinese, where the decisive effort appeared not to cross the goalline.
Juve showed no signs of their recent blip in the early stages, putting Inter under pressure right from the start and they almost went ahead in the second minute, with Fernando Llorente slicing just wide from 14 yards.
Inter failed to heed that warning, though, and Juve did break the deadlock just three minutes later.
Vidal flicked the ball past his Chile team-mate Gary Medel with an exquisite first touch inside the area, before sliding an inviting pass into the goalmouth for Tevez to steer home.
The goal appeared to spark Inter into life as the visitors started to dictate proceedings, though the champions never looked particularly worried at the back and remained a serious threat on the break.
And, unsurprisingly, Juve's next opportunity came at the end of a swift counterattack, as Samir Handanovic was forced to push a Vidal drive around the post.
Handanovic was at the centre of things again nine minutes before the break, blocking a Pogba effort from close range after the Frenchman delightfully slalomed between Juan Jesus and Andrea Ranocchia.
Mancini looked to give Inter extra attacking impetus by introducing Podolski for his debut eight minutes into the second half, though the German struggled to make an impact from the left.
Juve continued to look threatening on the break and they were somewhat unlucky not to get a penalty in the 58th minute, as Juan clumsily tripped Llorente.
Inter capitalised on that let-off, though, with Icardi beating Gianluigi Buffon from a tight angle after beating Giorgio Chiellini to Fredy Guarin's throughball.
Icardi agonisingly missed the target from point-blank range following Podolski's cross with nine minutes to go.
And the away side were relieved to settle for a draw as Kovacic's red card for a crude challenge on Stephan Lichtsteiner effectively ended their chances of victory, with Patrice Evra's deflected shot tipped away from danger by Handanovic.
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