The FA Cup defeat also condemned Liverpool to consecutive defeats for the first time this season, following their 2-1 league defeat to Manchester City, and it also meant that the side are now winless in the new calendar year.
Those results have raised some concerns, but Milner insists that he is not anxious about the group's ability to bounce back.
"I think the start of this season we weren't playing our best but we were getting results". That's how it sometimes starts - when you are really needed then it is only about if you are good enough and not how old you are.
That's stunning progress when you consider they are up against a club in Manchester City with a bottomless pit of money that has invested hundreds of millions to build a team that should be dominating English football. It's how you respond and how you bounce back.More news: New concession on backstop following furious row over Brexit vote
"It wasn't as easy to get the passing going in those tight areas as we would have liked but it was my mistake for the first goal and it cost us ultimately".
"It's disappointing but we have to make sure we recover and bounce back from this".
As quoted by the Liverpool Echo, Milner said: "I should have done better".More news: Manchester City v Burton Albion Betting Advice, 9 January 2019
"Ki coming on after a minute and playing like he did [was] absolutely outstanding", Milner, who made his professional debut for Leeds United at 16 years and 309 days, added. "You can see that straight away - his temperament, he's comfortable on the ball".
However, when asked whether their elimination from the competition helped them to prioritise their league ambition, Milner dismissed the suggestion, indicating that the side want to win "every game".
This is not a moment for Liverpool to hit the panic button, but there is no denying the Premier League trip to Brighton on Saturday has now taken on huge significance.
Liverpool's 1-0 defeat at Wolves followed their narrow Premier League loss against Manchester City last week. "The games keep coming thick and fast and hopefully we can get a few of the injured lads back as well".More news: Timberwolves 'rallying around' new head coach Ryan Saunders