The Reds claimed a dramatic 3-2 win over Atletico Madrid on Wednesday, maintaining their unbeaten start to the season.
And Milner hopes they can bring the same intensity into Sunday's clash with Manchester United.
On beating Atleti, Milner said: "Obviously there is work to do, nine points isn't enough and we need to make sure we do get over there and winning the group is our No.1 aim.
"But we're in a great position. It is a tough group – we know that – and we saw tonight what happens if you don't continue.
"We saw what happened in the first half when we conceded two goals and it turns 2-2. Now we're in a strong position in the group but we need to keep our foot down and keep this run of form going.
"We've got plenty of games coming up thick and fast, so we need to keep performing and putting in the performances."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!