Rodri struck in the 68th minute of Saturday night's final to see the Premier League champions and FA Cup winners complete a treble of trophies this season.
While it is the first time City has won European soccer's biggest club competition, it is the third time Pep Guardiola has lifted the trophy as a coach.
City triumphed despite losing inspirational midfielder Kevin De Bruyne to an injury in the first half.
Their success is the 10th treble of domestic league and Cup plus Champions Cup/League overall, and they are only the second English side to achieve it, following crosstown rivals Manchester United who did it in 1999.
"It is like a fairy tale, it can't get any better," City captain Ilkay Gündogan told streaming portal DAZN before receiving the trophy.
"We showed more courage in the second half. We are alert. You can't prevent everything against such a good team but in the end one goal made the difference."
The victory means the club finally achieved its ambition of reaching the summit of European soccer, 15 years after Abu Dhabi's ruling family transformed it into the richest teams in the world.
Owner Sheikh Mansour bin Zayed Al Nahyan was in attendance to see City's crowning moment. It was only the second time he has watched his team in person in 15 years.
Romelu Lukaku had the chance to score a late equaliser, but headed straight at Ederson from about four metres out.
City's winner came when Rodri collected Bernardo Silva's cutback and fired through a crowded penalty box.
The relief was unmistakeable as he raced towards City's fans and slid on his knees in celebration.
Inter almost evened the score within minutes of that goal when Federico Dimarco hit the bar from close range.
He then looked like turning in the rebound, but saw his shot come back off teammate Lukaku.
Assessing his own contribution in frank terms, goal hero Rodri said: "I wasn't good in the first half - I was playing s***, to be honest and then we talk about mentality, I just said to myself, 'You have to calm the situation'. At the end, I scored a goal. Unbelievable."
"Finals are like this," he added. "You expect to play well like always. The emotions are there, the nerves are there, but the lads, we compete until the end.
"Some of these guys are 20, 21, 22, some of them their first final. It's normal, but we compete like animals."
with PA
19°C