The Potters won 3-1 at Vicarage Road on Tuesday night to reach the fourth round of the competition for the third successive year.
Nick Powell put the visitors ahead before Ashley Fletcher levelled – only for Sam Clucas and Josh Tymon to strike late on and send Stoke through.
Michael O'Neill's side sit fifth in the table but lost at beleaguered Derby at the weekend and Chester wants to build on the win at Watford to get back on track in the league.
"It's not an easy place to come, Watford came up comfortably from the Championship last season," he told Stoke City's in-house video service.
"I think it's important for the momentum of the club in general. We got a disappointing result at the weekend but we started the season really positively and it's important we are keeping up victories and keep improving on our performances."
Chester has yet to feature in the Championship for Stoke this season and has been restricted to appearing in cup games.
But the former Hull and Aston Villa centre-back feels the fringe players who were given their chance as six changes made by O'Neill cannot have harmed their prospects of forcing their way in.
"Hopefully, that is the advantage of having such a good squad," he said when asked if the team had given O'Neill food for thought.
"We started well this season so that'll be up to him who he picks but we showed that if we are needed, we are already."
It was a bittersweet evening for Rob Elliot – the experienced goalkeeper on the losing side as he made his long-awaited Watford debut.
The 35-year-old joined from Newcastle in January and has had to be patient for his opportunities, which could once again be limited following the Hornets' cup exit.
"I'm really proud," he told the club's official website.
"I've been here a while and not really had a chance because the lads have done so well.
"I haven't played in a long time and I've really missed it. I haven't just come here to be number three, I try and push the lads everyday in training. I was hoping for a little cup run but that wasn't to be.
"Stoke came here with a game plan and stuck to it well, but there were so many positives to take.
"It's just very disappointing with the result at the end because we did more than enough to win the game."