Mohamed Salah put Liverpool ahead against the run of play three minutes before halftime and they then took command after the break with two quick goals from Darwin Nunez and Harvey Elliott to claim a convincing triumph.
The victory kept them three points behind leaders Napoli, who won at home against Rangers in Italy at the same time to make it five out of five in the group.
Napoli will be at Anfield next Tuesday to decide the top place when Liverpool would need victory by at least 3-0 to edge the Serie A leaders on the head-to-head tally between the two after losing 4-1 in Naples in September.
Ajax needed to win by two goals on Wednesday to keep up their slim Champions League hopes but will now be battling with Rangers in Glasgow next week for a Europa League spot.
The Dutch club have three points while Rangers are yet to open their account.
Steven Berghuis crashed an early effort against the upright as Ajax began in a lively fashion and with an unexpectedly physical approach that unsettled Liverpool’s defence.
It ensured some edgy exchanges between the players, indicative of what was at stake in the contest at the Johan Cruyff Arena.
The game turned suddenly, and firmly, in Liverpool’s favour when skipper Jordan Henderson’s outstanding pass, delivered with the outside of his boot, was met with an equally proficient finish from Salah who lofted the ball expertly over Ajax goalkeeper Remko Pasveer as he rushed out of goal.
It might have been 2-0 within 120 seconds as Nunez squandered an easy tap-in at the back post after an unselfish pass from Roberto Firmino, striking his effort against the upright.
Nunez made amends four minutes into the second half with a strong glancing header from a corner and the contest was over two minutes later when Elliot scored.
He thrashed a powerful shot into the roof of the net from an acute angle after being set up by a perfectly-weighted pass from Salah.