While the home side pulled one back through Ollie Watkins to make a game of it in the second half, City claimed the win to stay second in the standings on 32 points from 14 games, one behind leaders Chelsea.
Villa stayed 13th on 16 points after their first defeat under new manager Steven Gerrard following two successive wins.
City dominated the opening half and twice went close early on as Villa goalkeeper Emiliano Martinez parried fierce shots by Joao Cancelo and Rodri, with the home side barely able to break out of their own half.
The visitors took a 27th-minute lead through Dias, when the Portugal centre-back beat Martinez with a slightly deflected first-time shot from 20 metres after Raheem Sterling's cutback flashed across the penalty area.
If there was an element of luck involved in the opening goal, the second was sheer class as City broke away under pressure in their own penalty area and struck at the other end in a matter of seconds.
Cancelo threaded a short pass to Riyad Mahrez past two Villa forwards and the Algeria international released Gabriel Jesus on the right flank, with the striker lifting a cross for Silva to crash an unstoppable volley into the roof of the net.
Down but not out, Villa sprung back to life shortly after the break as Watkins found space in a crowded penalty area to drive in a half-volley inside the near post from 10 metres after he met a low Douglas Luiz corner.
The home side missed a gilt-edged chance to salvage a draw in the 75th minute when City keeper Ederson parried a sitter from substitute Carney Chukwuemeka and the visitors held on in the closing stages.
Dias was happy City were able to survive Villa's late onslaught and described Silva's goal as a morale-boosting moment.
"Very important win today against a team that are on fire under a new manager," Dias told Amazon Prime. "Very important to get the three points. Everyone fight a lot and we are very happy.
"Bernardo gave us a little bit of extra confidence into the second half. We have to give them credit for a good goal that changed the game. With the crowd they tried to make things different but our team was always in control."