Joe Thomson and Jamie Lindsay struck two goals in three minutes before a slack defensive clearance by the visitors was turned in by Liam Henderson just before the break.
Stuart Findlay's own goal shortly after the restart gave the Dutch side some hope, however Lewis Kidd's penalty secured the Scottish club's first three points in the competition.
A 6-2 win for Barcelona in Italy mean Celtic remain third in the table but they have closed the gap on AC Milan to three points.
Stevie Frail and John Kennedy retained the same starting XI which had narrowly lost to Barcelona in the last tie earlier in the month.
And similar to that match, the young Hoops began positively and confidently, immediately piling on the pressure.
Celtic were looking stronger with every attack and the hard work paid off just after the half-hour mark when Thomson opened the scoring for the Hoops.
Eoghan O'Connell stormed out of defence and threaded a ball through to the winger who showed a calm head to slot home his third goal in as many games.
It took only three minutes for the Bhoys to build on that strike, as Lindsay slammed an effort on target. Thomson bundled the ball into the back of the net, although it was already over the line.
And the young Hoops extended their lead to 3-0 on the 43rd minute through a poor clearance from Anwar El Ghazi, which hit off Henderson, causing the ball to fly into the Ajax goal.
Celtic were then hit with a setback on 50 minutes when Findlay turned the ball into his own net from an attempted clearance.
However, their three-goal advantage was be restored on 66 minutes with Henderson being fouled in the box, allowing Kidd to show great composure to score, and ensuring Celtic stay within touching distance of Milan.