After a first half of very little action, the diminutive Spaniard gave City the lead just two minutes before the break.
Kevin De Bruyne's corner from the right was won by Nicolas Otamendi and Silva was the first to react to hook home the loose ball from close range.
United appealed in vain for offside but the goal was a legitimate one.
Check out the strike for yourself below:
David Silva scores 1st goal of #ManchesterDerby
— José Villacreses (@jj_villac) 10 December 2017
Manchester United 0 - 1 Manchester Citypic.twitter.com/Skj3HpOBvU