The Reds fell behind midway through the first half as Villa forward Trezeguet capitalised on their poor defending at the far post as he finished from John McGinn's free-kick into the box.
The home side defended astutely for the following hour as Liverpool lacked the final ball to penetrate through their defence and grab an equaliser.
With time running out and Liverpool heading for their first Premier League defeat of the campaign, Jurgen Klopp brought on Alex Oxlade-Chamberlain, Divock Origi and Naby Keita as he looked to mix things up.
With a little over three minutes of normal time remaining, Mane sent in a brilliant cross from the right, finding Robertson at the far post. The Scotsman made no mistake with his header as he scored a crucial leveller for the visitors.
Klopp's side got an unlikely winner with just seconds remaining as Trent Alexander-Arnold's corner was turned in by Mane, who did superbly to find the back of the net with a glancing header.
Liverpool's impressive form continues as they maintain a six-point lead at the top of the Premier League table, with Manchester City to come next weekend in a huge clash at Anfield.
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