It took until the second half for the deadlock to be broken with the returning Vincent Kompany rising high to score his first goal since August 2015.
The result means that seven points now separate City from fifth place, and Guardiola acknowledged that it is another step towards their goal of qualifying for the Champions League.
Guardiola described his performance in last week's victory over Hull as "unbelieveable" and the Spaniard has now explained why he has been so impressed.
City were controlling the game and De Bruyne looked to add a second, firing well over before his deflected ball into the area fell to Silva, whose low effort hit a defender and span behind.
He also led City to their first clean sheet in five matches and Guardiola is hoping he can stay fit after muscle injuries that have reduced him to just 29 appearances in the last two seasons.
"That is why I am so, so glad for the fans we have because for the first time in my life I have experienced that". We were able finally to make 15, 20, 25 passes in a row.
Two soon became three following another De Bruyne assist, this time racing to the byline and floating the ball into a unsafe area for Aguero to power home his 11th goal in 11 outings.
"We knew this game was massive for us".
He said: "The problem when we concede goals is not the goalkeeper - it's not Claudio, it's not Willy (Caballero), it's all the team".
With six games remaining City have 64 points, one more than fourth-placed Liverpool, who visit West Bromwich Albion on Sunday. "From a set piece we scored a goal and they had to push up and after that we could get some players between the lines and counter attack and kill the game".
Despite Kompany's struggles, Guardiola believes his captain can still play a crucial role over the final weeks of the season.
Asked whether winning the cup or claiming a top four position to qualify for the Champions League was more important he said: "Why do I have to choose?" Not just the defensive part - he's helped us a lot - but he is doing all he can do to create chances. Shane Long, 28, will be a threat to Manchester City while Nathan Ward, 23, is emerging as a real talent for his team.