Mawson stunned the Reds shortly before half-time, giving the hosts the lead after a slow first half. Liverpool were the better side after the break but couldn't break down Swans' defence, despite a huge advantage in possession.
Here's a look at the teamsheet, via James Pearce of the Liverpool Echo:
Liverpool started with plenty of possession and had a great chance early, as Sadio Mane got in behind the defence but couldn't work the ball past Lukasz Fabianski. Replays suggested he may have been offside.
Roberto Firmino was the next man to go close, but the referee correctly brought his effort back for a handball.
Jordan Ayew was furious when he had the ball in the net after a corner had been given, and once again, replays suggested Swansea City had a right to be angry. On the other side of the pitch, Virgil van Dijk couldn't guide a header on target.
The Dutchman beautifully isolated Salah after 30 minutes with a pass over the top, but the Egyptian tried a difficult volley and couldn't keep the ball down. Football Whispers were impressed with how the hosts handled Salah:
Alex Oxlade-Chamberlain fired a shot right at Fabianski before Mawson gave the hosts a shock lead, taking advantage of a mistake from Van Dijk to power home after a corner.
Paddy Power couldn't help themselves:
Joel Matip and Andrew Robertson picked up late bookings, and Mane missed the final good chance of the half, firing a volley wide.
The Reds continued to press to start the second half, but chances were rare. Fabianski stepped ahead of Firmino to deny him an opportunity, and Matip risked a second booking for another bad foul on Ayew.
Fabianski continued his good showing with an excellent stop on Salah, who aimed a free-kick on target. Sportswriter Chris Wathan noted manager Jurgen Klopp lost his composure in the process:
Mane tried to force the issue with a dive after light contact with Kyle Naughton and was lucky not to be punished for simulation, per Tom Evans of the Liverpool Echo:
Emre Can and Matip both missed the mark with desperate drives, and Loris Karius had to be alert to deny Tom Carroll at the near post.
Klopp brought on Danny Ings late, and the Reds applied all kinds of pressure on Swansea, who could barely keep hold of the ball. Ings immediately made an impact with a lovely piece of skill, but once again, Fabianski kept his net clean.
Van Dijk missed with a wild volley and couldn't get past Fabianski with a header, and Salah went wide after good work from Adam Lallana. The final chance fell to Firmino, who somehow failed to equalise from a free header and instead found the woodwork.
Liverpool and Swansea will both be in action in the FA Cup next.