Ajax had dominated the early stages by continually finding gaps in behind Liverpool's full-backs, with Dusan Tadic wasting by far the hosts' best opportunity when firing straight at Trent Alexander-Arnold from 12 yards after Brian Brobbey's cutback.
After weathering an early storm from the hosts, Mohamed Salah broke the deadlock with three minutes of the first half remaining after latching onto Jordan Henderson's glorious through ball with the outside of his foot, before the Egyptian delicately chipped the onrushing Remko Pasveer.
Darwin Nunez should have doubled the visitors' lead two minutes later but he missed an open goal from a tight angle after Roberto Firmino had unselfishly squared the ball following a forward burst from Andrew Robertson.
However, the Scotland left-back's outswinging corner found Nunez four minutes after the break, with the Uruguay forward making amends for his earlier profligacy with a thumping header into the bottom corner to dampen any potential Ajax fightback before it had even began.
Liverpool's victory was sealed in the 52nd minute when Salah's perfectly weighed slide-rule pass found Harvey Elliott's forward run into the space which had been vacated by the Egyptian winger, with Elliott bagging for the second successive European match by slamming the ball into the roof of the net with his weaker right foot.
The Reds will now compete in the Champions League knockout stages for the sixth successive season under Jurgen Klopp's management, although they will need to beat Napoli by three or more goals at Anfield next week in order to leapfrog the Serie A leaders into top spot in Group A having been beaten 4-1 by them on matchday one.
Ajax, meanwhile, will almost certainly drop down to the Europa League having beaten rock-bottom Rangers 4-0 in the corresponding fixture, meaning last season's Europa League runners-up would need to win by four or more goals at Ibrox next week in order to finish third in the group.
With Liverpool trailing Arsenal by 12 points at the top of the Premier League, this competition undoubtedly represents Klopp's side's best opportunity of turning what has so far been an extremely difficult season into a glorious one should they go all of the way to the final next summer at the Ataturk Olympic Stadium in Istanbul, where of course they have plenty of history having famously beaten AC Milan from three goals down in 2005.