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."
Kompany urged City, who trail Premier League leaders Chelsea by six points, to use the victory as a catalyst and rediscover their best form.
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."
