Neil Lennon had insisted that his side could not afford to lose for the third straight game in Group H, and they managed to pick up their first three points of the campaign thanks to goals from James Forrest and Beram Kayal.
The breakthrough came on the stroke of half-time when Forrest scored from the penalty spot after Anthony Stokes had been brought down by Stefano Denswil.
Denswil’s night then went from bad to worse when he got a crucial deflection to Kayal’s long-range strike on 54 minutes to double Celtic’s advantage.
Nir Biton was shown a straight red card in the 88th minute for a late tackle on Thulani Serero but, despite Ajax’s best efforts and a wonderful stoppage-time goal for Lasse Schone, Celtic held firm.
Celtic were without suspended skipper Scott Brown and injured duo, Kris Commons and Adam Matthews, and it was Ajax who had the first chance on 10 minutes when a deflected cross from Daley Blind had Fraser Forster troubled before he cleared with his foot.
Moments later, Parkhead striker Teemu Pukki did not get enough on a fine cross from Mikael Lustig and the ball flew wide of the target.
As the first half unfolded, Ajax began to move forward with a bit more fluency but without posing the goal threat which the home side hinted at when they approached the Dutch target.
Just before the half-hour mark, however, the Parkhead side had a lucky escape.
A deep free-kick from Viktor Fischer from wide sailed over to the back post where Christian Poulsen sent a low volley crashing off the post with the Hoops defence scrambling the ball to safety.
Three minutes later, following a Charlie Mulgrew free-kick which had been cleared by the packed Dutch defence, Kayal was given a roar of approval by the home fans when he raced back to rob Fisher of the ball as he was ready to go clean through on goal.
Forrest’s penalty, which he slammed high past Jasper Cillessen, followed some delaying tactics from the Ajax keeper – with referee Bebek booking Poulsen and Joel Veltman and Hoops defender Virgil van Dijk.
In the 53rd minute Forster brilliantly denied Serero after the Ajax player had slipped through the Hoops defence with some ease.
Moments later the home side were two goals up. Georgios Samaras’ cross from the left was cleared to Kayal 20 yards from goal and his side-footed shot appeared to take a deflection off Denswil before nestling behind Cillessen.
The match opened up further. As Ajax desperately tried to get back into the game Kolbeinn Sigthorsson missed the target from 12 yards when he only had Forster to beat.
However, Celtic looked just as likely to get a third. Serero blasted over the bar in the 75th minute after getting the break of the ball and with that, the Dutch side’s chances all but disappeared.
Forster made another good save with five minutes remaining, this time from Blind’s drive, before Biton got a straight red for a hefty challenge on Serero.
Schone’s fine strike reduced the deficit but soon after the final whistle sounded – bringing a roar from Celtic fans, knowing their Champions League hopes have been rekindled.