Jurgen Klopp advanced from six consecutive group stages in the competition with Liverpool, a club record, after the Reds dominated on Thursday (AEDT) in Amsterdam.
Salah scored his sixth goal in his last four Champions League games with a 42nd-minute opener before Darwin Nunes and Harvey Elliott added quick-fire second-half strikes to seal victory.
Ajax will be relegated to the Europa League after avoiding a high-profile defeat at the hands of winless Rangers in the sixth round, while Liverpool must beat leaders Napoli by a narrow four goals to win Group A.
Ajax should have taken the lead in the third minute, but Steven Berghuis took a gilt-edged chance against the right post and the midfielder curled just wide soon after.
A determined Liverpool continued to have their backs against the wall before Andy Robertson converted a rare chance for the Reds, but Klopp’s visitors soon took the lead.
Jordan Henderson chipped in delicately for Salah to slot neatly past the onrushing Remko Pasveer and two minutes later Nunez inexplicably smashed the left post with a goal.
Núñez made amends after the break, heading Robertson’s corner off the right post, before Elliott fired into the roof of the net two minutes later.
Substitute Kenneth Taylor converted a presentable opportunity after the hour, although Ajax never threatened an unlikely comeback.