The Reds picked up their first silverware in six years following the penalty shootout win over Championship promotion-chasers Cardiff City.
Victory has given the England international a taste for success, and he says his team-mates are hungry for more this season.
That means navigating their way past Stoke City in the quarter-finals of the FA Cup and to push as hard as possible for a top-four finish.
"I think what the win does is makes you more confident and it gives you an incentive to go on and try and win and play in more finals," Downing told the club's official website.
"We have a great chance even though it will be a tough game against Stoke in the quarter-finals. And we are still going for a Champions League place with fourth position. There is a lot to play for and hopefully we can carry on doing well.
"There was a lot of new players coming into the team, and it should give us a lot of confidence and a chance to build on that. If we add a few more in the summer it will make the squad even stronger and we can push on further."
The former Middlesbrough trainee moved to Anfield from Aston Villa last summer in the pursuit of winning major trophies and he believes he, like the club, is improving all the time.
"I think I should score more goals. I've been unlucky a few times this season, missed a few chances. I was close on Sunday," he added.
"I just need to keep getting into the position and hopefully with a little more luck I'll get a few more.
"But I am enjoying life here. There is pressure coming to Liverpool, but you have to concentrate on your performances and try and do as best you can.
"I have enjoyed the last few weeks, the team has been doing well, I am doing fine and I want to kick on now until the end of the season. I want to help Liverpool push on."
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