Czech defender Ondrej Celustka scored the 76th-minute winner at San Siro for Trabzonspor, which only qualified for the competition because Turkish rival Fenerbahce was removed while the club is investigated for match-fixing.
United, last season's runner-up, rallied to draw 1-1 at Benfica thanks to veteran winger Ryan Giggs' stunning equalizer while Aleksandar Kolarov's free kick earned City a 1-1 draw at home to Napoli in its first match at Europe's elite level since 1968.
Elsewhere, nine-time winner Real Madrid opened its group campaign with a 1-0 victory at Dinamo Zagreb despite playing the last 17 minutes a man down after Marcelo's dismissal. German giant Bayern Munich, which has won the competition four times and will host next year's final, won 2-0 at Villarreal to take the early lead in Group A, which also contains Napoli and City.
CSKA Moscow rallied from two goals down to draw 2-2 with French champion Lille, while Ajax was held 0-0 by Lyon and Basel beat Romanian debutante Otelul Galati 2-1.
Inter's defeat piles the pressure on under-fire coach Gian Piero Gasperini, whose team was also beaten 4-3 at Palermo in their Serie A opener on Sunday -- conceding four goals for the first time in a league match in more than seven years.
"It's difficult, without a doubt," said Inter goalkeeper Julio Cesar. "In a moment like this we all have to stay together, work hard and find our way out of this situation."
Substitute Coutinho squandered Inter's best chance in the 88th minute, drawing one of a number of saves out of Trabzonspor goalkeeper Tolga Zengin.
In the other match in Group B, Lille powered into a two-goal lead by the 57th minute through striker Moussa Sow and midfielder Benoit Pedretti but was pegged back following a double by Seydou Doumbia, the Ivory Coast forward's second goal coming in the 90th minute.
Fierce rivals United and City have both won their opening four matches of the season in the Premier League, producing some breathtaking attacking football in the process, but they were brought back down to earth in Europe.
Oscar Cardozo's classy 24th-minute goal put Benfica ahead in the Group C match but the 37-year-old Giggs collected Antonio Valencia's pass, cut inside from the right and fired a left-foot drive from the edge of the area into the top corner for United's equalizer.
The visitors had goalkeeper Anders Lindegaard -- filling in for the rested David De Gea -- to thank for preserving a draw, with the Denmark international denying Nolito and Nicolas Gaitan late on.
"Going a goal down but coming back was the good part of the game for us," said United manager Alex Ferguson. "In the last 15 minutes they came into it and Anders made two terrific saves to keep us into it."
Basel is the group's early leader, with Alexander Frei's 84th-minute penalty securing the Swiss side all three points. Fabian Frei put Basel ahead, only for Marius Pena to level in the 58th for Otelul Galati.
Otelul defender Adrian Salageanu was shown a red card for bringing down Marco Streller to concede the penalty Frei tucked away. Basel also finished with 10 men when Benjamin Huggel received a second yellow card in the 87th.
Big-spending Man City's expensively assembled team was given a reality check by Napoli, which marked its first game at Europe's highest level since 1990 -- and the days of Argentina great Diego Maradona -- by going ahead through Edinson Cavani in the 64th.
Kolarov, who was later substituted after breaking his nose in an aerial collision, whipped home a free kick five minutes later to salvage a point for City, which had earlier hit the woodwork through Yaya Toure and Sergio Aguero.
Toni Kroos and substitute Rafinha were the goalscorers for Bayern Munich as the German powerhouse began its pursuit of a fifth European championship with a comfortable victory.
Angel Di Maria was the matchwinner for Real Madrid in the 53rd, the Argentina winger finishing off a flowing move that included Cristiano Ronaldo, Karim Benzema and Marcelo.
Brazil left back Marcelo was sent off in the 73rd when he earned a second yellow card for diving under Jerko Leko's challenge but Jose Mourinho's Madrid, playing in red jerseys for the first time in 38 years, held on.Suggest a correction