Midfielder Beram Kayal scored a second in the 54th minute with a deflected shot to give the home side a cushion.
In the 88th minute Hoops substitute Nir Biton, who replaced Mikael Lustig, was shown a straight red card by referee Ivan Bebek for his challenge on Thulani Serero before Ajax substitute Lasse Schone grabbed an injury-time consolation.
The Scottish champions have three valuable points from three fixtures ahead of their trip to Amsterdam in a fortnight's time.
The Hoops went into the game without suspended skipper Scott Brown and injured duo, Kris Commons and Adam Matthews, but they showed they had more than enough quality left for the Dutch side.
Ajax defender Christian Poulsen hit the post in 29 minutes but that aside, there was no real pressure on the Hoops goal in the opening period.
Fraser Forster made a crucial save from Serero just before Celtic got the second but the home side were well worth their win and can travel to Holland with confidence.
There was a feeling of do-or-die inside Celtic Park before the game with both sides knowing defeat would probably end their chances of progression in the tournament.
Despite losing skipper Scott Brown to suspension and Kris Commons and Adam Matthews to injury, the home side were boosted by the return of defenders Lustig and Emilio Izaguirre from a hip injury and illness respectively.
Forrest, who scored in the 1-1 Scottish Premiership draw with Hibernian at Easter Road on Saturday after coming on as substitute, came back into the team as the visitors started with Kolbeinn Sigthorsson leading their line.
It was a tentative start to the game by the Dutch champions who looked uncertain in possession as Celtic pressed them high up the pitch.
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 did not get enough on a fine cross from 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 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 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.
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 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.
COMMENT RULES: Comments that are judged to be defamatory, abusive or in bad taste are not acceptable and contributors who consistently fall below certain criteria will be permanently blacklisted. The moderator will not enter into debate with individual contributors and the moderator’s decision is final. It is Belfast Telegraph policy to close comments on court cases, tribunals and active legal investigations. We may also close comments on articles which are being targeted for abuse. Problems with commenting? firstname.lastname@example.org