In a match that witnessed several tense moments, Hugo Boumous scored the winner for Bagan in the 67th minute.
Kerala are placed in Group A along with hosts Goa, Jammu & Kashmir, Chhattisgarh and Gujarat.
East Bengal failed to capitalise on the home advantage at the Salt Lake Stadium.
Jammu & Kashmir, Chhattisgarh and Gujarat are the other teams in Group A.
Mumbai City won 2-1 with all three goals scored in the first half.
Liverpool won 3-1 while Arsenal were held 2-2 by Spurs.
India will play Saudi Arabia in the pre-quarterfinals.
Goals in either halves by Jerry Mawihmingthanga and Diego Mauricio helped Odisha defeat Chennaiyin 2-0 to start their campaign on a winning note.
In all, referee Anthony Taylor doled out 11 yellow cards along with Rodri's red, with seven of the yellows coming in a bad-tempered second half as Forest tried to leverage their extra man, but to no avail.
Bengaluru FC striker Ryan Williams had allegedly racially abused Blasters defender Aiban Dohling during the inaugural match of ISL Season 10.