City were frustrated by their recently promoted opponents in the first-half, with Sterling spurning the only opportunity as Heurelho Gomes made the save for Watford.
However, the £44million summer signing moved into a more attacking position after the break, with the move paying dividends as he struck his first goal in City colours in the second half.
Fernandinho added a late second for the league leaders, who are now on four wins from four.
While Pellegrini was quick to laud the efforts of his side, he reserved particular praise for former Liverpool man Sterling.
“Raheem has played very well in all the games,” he told Sky Sports. “Of course he wants to score – it was good for him to do it. He had another chance in the first half, but I think he worked very hard in the whole game.
“I’m sure he will score more goals.”
Pellegrini was happy with his side’s patience in securing the victory, but would not be drawn on the eight-point gap his side have opened up on champions Chelsea at this early stage.
“It was not frustrating in the first half. You can’t expect to win the game inside 45 minutes against a team who defends well. We made some changes and we created more chances and won the game.
“It is difficult to ask for better – we have 12 points, nobody has scored a goal [against us] and we are scoring a lot of goals.
“But it’s also important to go game by game, to understand that we are just starting the season.
It’s a long season and the Premier League is very difficult in every game.
“Remember last season we were eight points behind Chelsea, then by January we were exactly the same at the top of the table.”
On the potential transfer of Kevin De Bruyne, with the Wolfsburg midfielder reportedly travelling to Manchester this weekend ahead of a deal for upwards of £50million, the Chilean added: “I’ll give you the same answer as always. When things are finished I will talk about it.”