Inter were six minutes away from a first Serie A victory at Stadio Olimpico since 2008 after Ivan Perisic's goal early in the second half on Saturday, before Nainggolan netted with a sliding effort after Edin Dzeko had mishit a shot into his path.
The Belgian midfielder's strike meant Roma, whose eight-match winning streak in the league was ended, remain five points clear of Inter in the third and final Champions League qualification place, but Mancini is still hopeful of catching them.
"The team put in an excellent performance, which builds on our recent displays," he said.
"There are still so many points up for grabs. If we finish on level points with Roma, we have got the advantage of a better head-to-head record.
"It's still open, we have got eight more games and a long way to go. We need to win every game from now until the end. If we do that, I'm sure we will get third spot.
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"There are three teams in front of us [Juventus, Napoli and Roma] who began the season with their eyes on the Scudetto, we are still in the mix and we will keep on pushing for the third-place finish.
"We did really well until Roma threw on Edin Dzeko. After that we found it tough as they switched to long balls.
"Clearly if you concede an equaliser that late there is going to be regret and perhaps we should have scored the second goal.
"We are frustrated that we had to do without three strikers – if we had at least one of Mauro Icardi, Rodrigo Palacio or Stevan Jovetic, we would probably have won the game.
"Roma stepped on the accelerator in the final 10 minutes and they scored on the least well-worked move because Dzeko scuffed that ball and it turned into an assist."
Fifth-placed Fiorentina are also in the mix to finish in the top three and they sit one point behind Inter with a game in hand against struggling Frosinone today.