The favourites would have been 2-0 down had Kompany not denied the Italian with a superb tackle just before the break.
In the end it took two "wonder-strikes", as Gus Poyet called them, to put City in front.
Toure curled a dipping shot over Vito Mannone from 30 yards to level and Nasri clipped a sweet half-volley just inside the Sunderland stopper's far post from Aleksander Kolarov's pass.
Sunderland manager Poyet was stunned by the quality of both goals.
Gus Poyet: "I am proud of my team.
"For 45 minutes we were more than decent but then Toure scores from 35 yards in the top corner. Maybe if we had two goalkeepers we would have stood a chance."
"It's not nice when you are on the losing side.
"I hate it but someone needs to win. It's a final.
"The difference was two special goals. We didn't take our opportunities and then it was just like 'bang-bang' and it's over.
"We tried our best. Now we just need to try to stay in the Barclays Premier League.
"On another day we would have got a foul on Seb Larsson in the run-up to the goal.
"If someone puts the studs into your foot when you clear the ball it's a foul.
"But I am not going to be a hypocrite like most to blame the referee because Yaya Toure is an outstanding player, but it was a foul.
"It was a very easy foul to give. Maybe you can call the ref and ask him, he will have an excuse. They always have."
the score. It was important for the players to be calm and they were."