Cilic ended a three-match losing streak. He won in Moscow in 2014 and 2015.
Dzumhur won here in 2017. But this weekend he came through qualifying to play his first main draw match on tour since July.
Cilic earned his 19th tour-level title on Stuttgart grass in June and, at No. 6, was the only men's seed in action on Monday. The leading seeds were all locals; Andrey Rublev, champion of the last Kremlin Cup in 2019, Aslan Karatsev, and Karen Khachanov, the 2018 champ.
Tomljanovic set up a second-round match with top-seeded Aryna Sabalenka, who received a wild card to play her first tournament since reaching the US Open semifinals.
The Aussie has a win over Sabalenka in her past and brings some form to their fifth career matchup after downing Garbine Muguruza last week at Indian Wells. Muguruza is at the other end of the draw this week as the second seed.
Former champion and regular threat Anastasia Pavlyuchenkova had a bye as one of the top four seeds, and learned her second-round match will be against Bernarda Pera, the American who defeated Aleksandra Krunic of Serbia 7-6 (3), 6-3.