James Forrest scored from the spot at the end of the first half after Ajax defender Stefano Denswil had clattered Irish striker Anthony Stokes.
Midfielder Beram Kayal scored a second in the 54th minute with a deflected shot.
In the 88th minute Hoops substitute Nir Biton was shown a straight red card for his challenge on Thulani Serero before Ajax sub Lasse Schone grabbed an injury-time consolation.
Celtic boss Neil Lennon was delighted with his side’s performance.
“We were anxious at times in the first half. But as the second half went on we got better, and thoroughly deserved to win the game.”
Despite losing skipper Scott Brown to suspension and Kris Commons and Adam Matthews to injury, Celtic were boosted by the return of defenders Lustig and Emilio Izaguirre from a hip injury and illness respectively, and the Bhoys started well.
However, in the 10th minute, in Ajax’s first real foray, a deflected cross from Daley Blind had Forster troubled before he cleared with his foot.
Moments later, Parkhead striker Teemu Pukki could not get enough on a fine cross from Lustig.
Just before the half-hour, the Parkhead side had a lucky escape as Poulsen sent a low volley crashing off the post with the Hoops defence scrambling the ball to safety.
Three minutes later, Kayal was given a roar of approval by the home fans when he raced back to rob Fisher as he was ready to go clean through.
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, and moments later the home side were two goals up. Samaras’s cross from the left was cleared to Kayal 20 yards from goal and his side-footed shot deflected off Denswil before nestling behind Cillessen.
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 hopes of progression have been rekindled.
“It leaves us in a very good position,” said Lennon. “We have to get a result in Ajax if we can. It’ll be difficult but on tonight’s evidence not beyond us. It’s opened up the group and was a massive win.”
CELTIC: Forster, Ambrose, Lustig (Biton 77), van Dijk, Mulgrew, Samaras, Pukki (Balde 90), Kayal (Ledley 70), Izaguirre, Stokes, Forrest.
AJAX: Cillessen, Van Rhijn (Schone 80), Veltman, Denswil, Blind, De Jong, Poulsen (Boilesen 68), Serero, Andersen, Sigthorsson, Fischer (De Sa 72).
Referee: Ivan Bebek (Croatia).