But it was Fati who stole the show just a minute after coming off the bench, exchanging passes with Suarez on the edge of the area before arrowing a low shot in off the post.
Messi has been left out of the game at San Siro, with Barca having already qualified for the last 16 as Group F winners.
Romelu Lukaku equalised with a low 25-yard strike but Fati's late victor consigned Inter to the Europa League.
Lukaku was otherwise wasteful in front of goal, however, missing a particularly fine opportunity early in the second period, before the visitors sealed Inter's fate late on through 17-year-old Fati, dropping the Italians into the Europa League.More news: Ashley Benson Confirms She's Still Dating GF Cara Delevingne
Inter now occupy second place in Group F, ahead of Borussia Dortmund on goal difference, and know that anything other than a win could see the German side pip them to the knockout stages of the Champions League.
In the process, Fati broke a record previously held by Peter Ofori-Quaye, who scored for Olympiacos aged 17 years and 194 days in 1997.
'But Inter have so many good players, Lautaro, Lukaku, [Milan] Skriniar...'More news: A Fully Loaded Apple Mac Pro Will Cost You $52,600
Martinez had the ball in the net on 80 minutes, but again it was ruled out for Lukaku's offside position.
Barcelona have become only the second side to avoid defeat in 20 consecutive Champions League group stage games (won 14, drew six), after Real Madrid between 2012 and 2017 (30 matches).
Inter Milan host FC Barcelona in the pick of tonight's UEFA Champions League action, with the Serie A club needing a win to guarantee progression to the knockout stages of the competition.More news: Heavy gunfire reported in Jersey City neighborhood