The Reds progressed to the fourth round of the FA Cup after beating Aston Villa's youth side at Villa Park after a COVID-19 outbreak amongst the senior players.
However, three Liverpool goals inside a five-minute spell saw the Reds take an unassailable 4-1 lead as they cruised to victory in the final stages.
The boss would've certainly breathed a sigh of relief after the result, with his side finally getting back to winning ways after two draws and a defeat in their previous three outings.
Having thrashed Crystal Palace 7-0 in mid-December, the goals had dried up for the Reds before tonight and Milner believes it's Liverpool's final ball that has been the missing piece of the puzzle.
"When you're saying chances, we're getting into areas but not getting the right final ball, the right run," Milner told BT Sport after the game.
"That's been the story of the last few games as well, so it's good for us to work on that and we need to keep improving.
"You can never fault the effort of us, but at times things aren't quite coming off at the moment, and we need to keep working hard.
"You saw the attitude of the boys – they're pushing hard to put things right."
The return of Thiago should hugely help Liverpool with their final ball issue, with the Spaniard once again showing his world-class abilities after coming on as a substitute at Villa Park.
Fellow substitute Xherdan Shaqiri was involved in two goals after coming on and his reemergence could help the Reds freshen things up at a time when their attacking verve seems stagnant.
Milner went on to praise the opposition, labelling them a "good set of lads" who "worked hard for each other" as he said the Reds needed to lift themselves at half-time after seeing their opener cancelled out minutes before the break.
"I think obviously we needed to do some things better. I think we started pretty well with the counter-pressing and pressed on the ball, but then maybe didn't get it right," Milner said.
"Especially for the goal, [we] let them out of a couple of situations in the corner where they shouldn't have got out.
"They're good players; they're at Aston Villa for a reason. Every single time you're not quite there, you're going to get punished. They took the goal and we had to lift ourselves again."
The Reds now have nine days to prepare for Manchester United's visit of Anfield, with Ole Gunnar Solskjaer's side having the chance to go above the Reds at the top of the Premier League table before their clash.