Malcom's 83rd-minute strike looked to be sealing all three points for Barcelona, but Icardi came up with a leveller four minutes later to ensure that the points would be shared.
The result has seen Barcelona remain top of Group B, now on 10 points, with Inter second on seven points with two group matches left to play.
Luis Suarez was next to try his luck for the visitors after a poor kick from Handanovic set up a shooting chance for the striker, but his effort just missed the crossbar.
Inter's first serious chance of the match came in the 19th minute when Kwadwo Asamoah arrived inside the Barcelona box to meet a low cross, but the full-back sent the ball too high.
Dembele had a goal-bound strike blocked by Luis Suarez just before the 30-minute mark, before the latter fired wide of the far post with Inter standing and watching from a half-cleared corner.
Barcelona certainly had the best of what was an open first 45 minutes, and they had another opportunity to make the breakthrough just before the break, but Luis Suarez was denied by Handanovic as the two teams headed down the tunnel at 0-0.
Stefan de Vrij had the first half-chance of the second period when he met a Perisic cross in the 50th minute, but the centre-back's header dropped over the Barcelona goal, before Handanovic kept out a strike from Philippe Coutinho down the other end.
Clement Lenglet headed over the Inter goal just before the hour as Barcelona came close once again, before Handanovic made a smart save to keep out an effort from Rakitic.
Handanovic remained busy as he was forced to keep out a strike from Coutinho in the 62nd minute, before Matteo Politano headed a wonderful Inter opportunity wide of the post in the period that followed.
Another smart Handanovic save denied Dembele in the 67th minute, but Inter were putting the pressure on down the other end, with Borja Valero and Perisic both having strikes deflected wide.
Arturo Vidal and Malcom both came off the Barcelona bench in the period that followed, and the latter enjoyed his biggest moment for the club in the 83rd minute.
Malcom has had to be patient since arriving at Camp Nou, but he sent Barcelona ahead with a low strike which squirmed through the arms of Handanovic.
Inter were level four minutes later, however, when Icardi picked up a loose ball inside the Barcelona box before turning and finding the back of the net with an excellent finish.
The result has left Inter three points clear of third-placed Tottenham Hotspur with two games left in Group B.
INTER MILAN (4-2-3-1): Handanovic; Vrsaljko, De Vrij, Skriniar, Asamoah; Vecino, Brozovic (Martinez 85'); Politano (Candreva 81'), Nainggolan (Valero 63'), Perisic; Icardi
BARCELONA (4-3-3): Ter Stegen; Roberto, Pique, Lenglet, Alba; Arthur (Vidal 74'), Busquets; Rakitic; Dembele (Malcom 81'), Suarez, Coutinho