Downing joined the Reds in the summer from Aston Villa for a fee believed to be around £20million, and it was the midfielder who struck the decisive goal in the 2-1 win from just inside the box in the 57th minute.
Kenny Dalglish's side will now face the winners of the replay between Sunderland and Everton in the semi-finals with Downing admitting Sunday's effort had been his best moment in a Liverpool shirt.
"I'm absolutely delighted. It was a tough game, but the main thing is we got through. Now we have another trip to Wembley," he told Liverpoolfc.tv.
"I think it's got to be the highlight [of my Liverpool career so far]. I've enjoyed some big games, and the [Carling Cup] final was great.
"It was a tough game for us, but the way we got the goal, and for me to score at Anfield, was a great feeling and I'm glad it won us the game."
Luis Suarez had put the hosts ahead midway through the first half and although former Reds striker Peter Crouch equalised quickly, Downing's second goal of the season knocked out last year's runners-up.
"It was a great goal from Luis. We've seen in training between Maxi and Luis they've got an understanding. It was a great one-two and I think he used the defender as a guide to bend the ball around the 'keeper and I think that's what beat him. It was a great finish and goal for Luis.
"I think we deserved to win. We were the only team going forward and creating. That's no disrespect to Stoke - a lot of teams come here and sit behind the ball.
"We have to be patient at times and I can understand fans getting frustrated when it's not going well, but if they bear with us we've got the players to open up defences - and we proved that in the end."
Meanwhile, Liverpool remain confident the Football Association will make the right decision and ensure their semi-final does not clash with the 23rd anniversary of the Hillsborough disaster, where 96 fans died, on April 15.
Liverpool have always had special dispensation to avoid matches on that date.