Dzeko actually scored his first Inter goal before the deal was confirmed, netting in a 3-0 friendly win over Dynamo Kyiv behind closed doors.
After the game, Inter revealed the 35-year-old Bosnia-Herzegovina captain has joined on a two-year deal.
They have also added Denzel Dumfries from PSV Eindhoven on a four-year deal.
Defender Dumfries produced some eye-catching displays as an attacking full-back for the Netherlands during Euro 2020.
Inter start their Serie A title defence next Saturday against Genoa.
Dzeko – who has scored 119 goals in 260 appearances for Roma – is a direct replacement for Lukaku, who joined Chelsea for £97.5m.
He won the 2008-09 Bundesliga title with Wolfsburg and the Premier League with City in 2011-12 and 2013-14.
Roma boss Jose Mourinho is interested in signing Chelsea striker Tammy Abraham to replace Dzeko.
