Egypt - African champions a record seven times - had not reached a World Cup since 1990, but the 25-year-old scored from the penalty spot deep into stoppage time with the scores level at 1-1.
Shortly after winning the award, Salah said: "I am very happy to win this award, and you feel like you had a great year, so I'm very happy".
Keita, who is set to join Mane and Salah at Liverpool this summer and will replace Salah as Liverpool's most expensive player ever, finished fifth.
Salah won the highest vote in the popular referendum on the official website ahead of Gabon's Pierre Emeric Obamyang, Guinea's Nabi Keita, Senegal's Sadio Mane and Niger's Victor Moses.
He carried AS Roma (2nd in Italian Serie A), in the first half of the year, scoring 15 goals and delivering 11 assists. "I am a really lucky person because I've had the opportunity to work with a few fantastic, outstandingly good players - and now I am happy I have it with Mo", Klopp told Liverpool's official website. "The manager has helped me to settle into the team".
He added, "I want to be the best Egyptian ever so I work hard".
"I would like to thank my Liverpool team-mates and I also had a good season with Roma so I have to thank my team-mates there and my team-mates in the national team". "It's a big pleasure, to be honest, to work with him".