Edin Dzeko gave the Giallorossi the lead early in the first half, as the home side poured on the pressure.
Roma would go onto hit the post three times in search of a second goal but would pay for that misfortune - as Mauro Icardi scored twice in the final 25 minutes to turn the match around.
Matias Vecino added the clincher in the final five minutes to seal the three points.
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New coach Eusebio Di Francesco had one big decision to make - who to use to replace the injured Bruno Peres, whose absence meant Roma were without a natural right-back option in the squad.
In the end he opted to move Juan Jesus into the role, with Federico Fazio slotting into central defence alongside Kostas Manolas.
Otherwise the side remained unchanged from the one that beat Atalanta on the opening weekend.
4' - Early pressure from the Giallorossi. Fazio meets a low corner but the ball bobbles wide of the near post...
12' - A first opening for Inter - they pour forward but Icardi cannot put his shot on target...
14' - POST! CLOSE! A deep cross is half-cleared and Kolarov thrashes a shot at goal from range - but it hits the woodwork!
14' - ROMA LEAD! DZEKO! What a strike! Nainggolan clips a perfect ball in for the striker, who chests it down and then volleys past Handanovic! 1-0!
18' - Candreva goes into the book for a poor challenge on Perotti. Roma win a corner as Defrel's shot is deflected over...
22' - Juan Jesus is booked for a tackle on Perisic down the Roma right flank. The resulting free-kick comes to nothing...
27' - Dzeko, full of confidence, forces a save from Handanovic with a curling shot from 25 yards...
42' - POST AGAIN! This time it is Radja who crashes a shot against the frame with a thumping low drive!
43' - SAVE! Inter counter through Candreva and eventually find Icardi, but Alisson stops his stinging shot!
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50' - Pressure from the hosts to start the half. Perotti curls over from 25 yards out.
57' - Perotti goes down under pressure from Skriniar in the box - he wants a penalty, but the referee waves it away...
65' - POST ... AGAIN! THIRD TIME! Roma put together a brilliant move, then Perotti strikes the woodwork from 25 yards out!
67' - Goal. Inter are level. Candreva feeds Icardi and the striker drives home inside the box. 1-1.
68' - Almost an instant reply from Roma but El Shaarawy's flick over Handanovic is cleared away off the line!
74' - What a game this is! Roma counter quickly but Perotti's ball is just too far beyond Radja, who can't finish it off!
77' - Goal. Inter ahead. Icardi gets his second with a close-range finish. 1-2.
79' - Alisson makes a good one-on-one save to deny Icardi.
82' - Great chance for Roma - it falls to Radja but his shot goes over the bar!
87' - Goal for Inter. On the counter, Perisic finds Vecino for a close-range finish. 1-3.
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"Unfortunately we were guilty of not putting the game to bed earlier on and we made a few mistakes on the first goal we conceded. It’s a shame because the team’s performance was almost flawless."We bossed the game, kept the ball well and we were switched on at the back. We were naive on their equaliser which changed the game. It basically comes down to the fact that we didn’t score despite a host of chances.
- Eusebio Di Francesco
The players now have a brief hiatus for the international break - with those not away with their respective countries likely to be involved as the Giallorossi host Chapecoense in a prestigious friendly at the Stadio Olimpico on Friday 1 September.
The next league game for Di Francesco and his men is away from home on September 9 - as they travel to face Sampdoria in Liguria.
Goalscorers: Dzeko (14'); Icardi (67', 77'), Vecino (87')
Roma: Alisson; Juan Jesus, Manolas, Fazio, Kolarov; Nainggolan, De Rossi (Cengiz), Strootman; Defrel (El Shaarawy), Dzeko, Perotti (Tumminello).
Inter: Handanovic; Nagotomo, Skriniar, Miranda, D'Ambrosio; Vecino, Gagliardini (J. Mario); Perisic, Valero, Candreva; Icardi.
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