Salah's second strike marked his 90th Premier League goal and put him in pole position in the race for the 2020-21 Golden Boot.
"It's a good win for us -- they are a tough team," Salah said after the game.
"We need to win every game now. I knew there was a big responsibility on me and I just try -- every game, not just now -- to help the team.
"City were struggling in the beginning, then we were struggling three or four games ago. Now we need to keep winning if we want to win the league. It's been a really big week."
The first half was far from inspiring with both teams managing just one shot on target apiece.
Salah danced around around the West Ham defence inside the box before curling a left-footed shot around the players and into the top right corner.
Just over 10 minutes later Liverpool broke through Trent Alexander-Arnold and Xherdan Shaqiri after a West Ham corner. With Liverpool outnumbering the defence, Salah deftly controlled the ball before hitting it past Lukasz Fabianski.
Dawson pulled one goal back for West Ham following a corner on 86 minutes.