The Reds gave themselves a two-goal advantage after an own goal from ex-West Ham defender Harvard Nordveit added to Trent Alexander Arnold's first-half goal.
If Liverpool enter the Champions League group stage and finish third in their group, they will then enter the Europa League for the Round of 32.
With 74 minutes gone the hosts got the second goal they had been threatening, Milner getting down the right and his attempted cross taking a deflection off substitute Nordtveit and looping into the far corner.
However, the club has revealed that they will not allow the player leave before the first Premier League game against Watford on Sunday and are prepared to reject even bigger bids for their playmaker.
Alexander-Arnold was making his European debut because of injury to regular right-back England global Nathaniel Clyne but justified his inclusion with a strong performance in both attack and defence in addition to the 35th-minute goal.
The Reds face the Bundesliga side in their first leg qualifier with Jurgen Klopp looking to gain an advantage going into next week's second leg at Anfield. "We gambled a little too much on the wing - it was a little risky in the first half".
Unfortunately for Hoffenheim, the free kick which was awarded before Milner received the ball on the break, was taken while the ball was still moving.
Young Boys dominated the second half against CSKA yet the Swiss ended up losing after an nearly comical own goal.
An aimless ball forward was headed into the air by Young Boys center back Kasim Nuhu, who compounded that clumsy touch by heading the ball for a second time, this time over goalkeeper David von Ballmoos and over the line. Sporting Lisbon and Steaua Bucharest drew 0-0.
Liverpool reached the playoff round by finishing fourth in the English Premier League, narrowly clinching the spot over surging Arsenal on the final day of the 2016-17 season.