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Liverpool, beaten 2-0 by Napoli in their opening game, quickly took charge, taking the lead in the ninth minute with a beautifully worked goal from Sadio Mane.
Left back Andy Robertson made it 2-0 in the 25th minute and it looked game over when Mohamed Salah added the third 11 minutes later.
The Austrians, however, gave themselves a lifeline through a magnificently-taken goal from South Korean forward Hwang Hee-chan six minutes before the break.
A superb 56th-minute volley from Japanese striker Takumi Minamino, after a cross from the left by Hwang, reduced the deficit further before Anfield was stunned four minutes later.
The impressive Minamino slipped the ball across the face of the goal and 19-year-old substitute Erling Haaland tapped in.
Yet Salah secured the points for Liverpool in the 69th minute, the Egyptian firing home his second goal after a header down from the always-influential Roberto Firmino.
The win left Liverpool second in the group on three points, one behind leaders Napoli who were held to a 0-0 draw at Genk earlier on Wednesday. Salzburg also have three points after thrashing the Belgians 6-2 in their opening game.