Manuel Pellegrini's men headed into the clash on the back of a three-game winless run in all competitions.
Yet having been aided by Chris Smalling's silly dismissal in the opening half, City secured a fourth successive league derby win for the first time since 1970 thanks to Sergio Aguero's goal.
Toure said: "We proved we are a strong team. We needed to bounce back and I think we played very well."
There was a palpable sense of relief among the home fans in the Etihad Stadium at full-time, with City holding off a late United rally to seal the points.
Skipper Vincent Kompany added: "There was no real explanation for it (the nerves). They played well towards the end.
"We had most of the chances but, if you miss a couple, all of a sudden you lose control of the game.
"Ultimately we stood strong, got a clean sheet and a good derby win. A derby win is massive.
"For this game, the league and everything else doesn't matter. Winning this game gives us an edge in the city again."
He said: "Gradually we'll get there again. We have a certain way of playing and sometimes if players are off form it's difficult.
"We're working hard every day to try to make sure we are a better team than last year."
