The Reds made a quick start pressing high up the pitch, with Divock Origi having the best of the early chances.
Fifteen minutes in, Jordan Henderson came close to giving Liverpool the lead when he surged into the penalty area from midfield, but goalkeeper Eugen Galekovic came off his line in time to gather the ball before the captain could get a touch.
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As Liverpool kept knocking on the door, Origi forced a save from Galekovic by unleashing a shot from inside the box, but it was too weak to cause trouble.
At around the half-hour mark, Jordon Ibe got involved by delivering a low drive towards goal, but Adelaide's shot-stopper was able to make a routine save to deny the Liverpool youngster.
Milner tried to add to his goalscoring efforts from last week in Brisbane by bringing the ball under control on the edge of the box and executing a shot at goal, but Galekovic palmed the ball away.
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After the half-time break, Dejan Lovren squandered a chance to put Liverpool ahead when he headed over the bar from close range, but the deadlock was finally broken by Milner.
A well-executed pass from Ibe was delivered into the box and Milner, who joined the Reds from Manchester City on a free transfer last month, was there to knock the ball into the back of the net from a few yards out.
Minutes later, Milner fumbled when he was caught on the ball by Tarek Elrich, who unleashed a 25-yard shot, but just edged past the post for Adelaide.
By the 80-minute mark, the visitors should have been two goals to the good when Ings met Henderson's free kick with a header, but his effort glided over the bar.
The former Burnley man came good eventually, though, as he managed to find the back of the net by rounding the keeper and slotting the ball through after latching on to Nathaniel Clyne's pass.
Liverpool will face a Malaysia XI when they return to Asia at the weekend.