The winger netted with a right-footed effort from distance as the Anfield side posted a 1-0 win which sent them top of Group A.
And, after struggling to earn a first-team berth in the Premier League so far this season, the 28-year-old hopes his goal has improved his chances of featuring in the future.
"I’ve said before the only way you can prove anything to the manager is out on the field – to play well and get goals or assists," he told reporters.
"So I’m happy but obviously I don’t know what the team is going to be for Sunday but hopefully I’ve given [Brendan Rodgers] a little bit of food for thought."
The former Middlesbrough wideman went on to underline the importnace of notching three points against Guus Hiddink's side and taking control of the group.
"It was a good point as well because they made it hard for us," he added.
"We knew it was probably going to need a long-range strike to win it because we’ve tried everything and they were breaking us down. It was frustrating at times but it was a good night for us in the end.
"We had to do our bit first but we got a good result and it was a perfect night for us.
|13/2||Stewart Downing is 13/2 with bet365 to score at anytime against Everton on Sunday
Downing also insisted he was happy to move to left-back after starting on the wing following Glen Johnson's withdrawal at half-time.
"I enjoyed it, I’ve enjoyed it when I’ve played there before," he said.
"Glen was struggling with a bit of a niggle at half-time so we didn’t want to risk him and I’ll go to left-back no problem. That’s where I got the goal from so a good night all round."
The England international also looked ahead to Sunday's fixture at Goodison Park, claiming the Reds are full of confidence going into the game.
"[We've had] back-to-back wins so confidence is high but form goes out of the window in a derby, it’s a big game," he continued.
"At Everton the crowd is going to be right behind them but we look forward to it, we won there last year so hopefully we can do that again.
"For our league position [getting three points] will be important. Everton have got off to a good start this season, have got some good players and seem to have the momentum a little bit but we’re fairly confident we can get a result there."
Downing's team-mate Brad Jones also expressed delight with Liverpool's performance, hailing the manner in which the hosts controlled their opponents.
"I think we did well. They could have been a dangerous team but we pretty much nullified their attacking threat and controlled the game," he said.
"The goal that came was a brilliant strike from Stewie [Downing]; we maybe could have created a little bit more but the fact that we controlled the game and defended so well took out their attack.
"[Rodgers] was happy. The midfield did really well, the likes of Nuri Sahin and Stevie – Jonjo was more on the attacking side - really controlled the centre of the park.
"The boys at the back didn’t give away any chances which is the aim of the game; we don’t want to be giving anyone cheap chances on goal and we did that tonight.
"I’m happy with the clean sheet but I’m more happy to get another game. It’s even better when the team defend like that and you don’t have anything to do."
The Australian goalkeeper, who has featured in the Reds' last two games due to Pepe Reina's absence with injury, also revealed his excitement at the possibility of starting against the Toffees.
"It’d be amazing [to play against Everton] but we’ll see what happens. I think Pepe is close to full training so we’ll see what happens and see what the manager does," he added.
"It’s always going to be a big game and Everton have been flying this season early doors so they’re going to be dangerous and obviously up for the game. It’ll be a tough test but one that we’re relishing.
"[We've given] two good performances – solid performances rather than anything flamboyant – but that’s probably better to be honest. There’s a good team work ethic, we’ve got results by doing the right things and that’s a big plus."