Once again, Celtic can dare to dream in the Champions League after a victory that boosted their manager Neil Lennon's hopes of guiding his side to the knock-out phase for a second season in succession.
Before last night the omens had not looked good. Apart from the loss of Victor Wanyama and Gary Hooper in the summer sales, Celtic had to contend with the enforced absences of their captain, Scott Brown, Adam Matthews and Kris Commons for the visit of Ajax, with a win vital after two defeats without scoring.
But goals from James Forrest and Beram Kayal secured a deserved victory and not even the late sending off of Nir Biton or Lasse Schöne's consolation goal for Ajax could temper Celtic's delight.
"We are in an excellent position now, where qualification is not beyond us," Lennon said. "This was a massive win in the context of the group. It was a heroic effort."
Frank de Boer, Lennon's Ajax counterpart, was hardly gracious. "Celtic did not create anything but scored two goals," he said. "We dominated the game."
De Boer's focus would be better placed on Stefano Denswil, whose foul on Anthony Stokes handed Celtic a penalty from which they took the lead.
The frantic early exchanges resembled a knock-out tie. Celtic enjoyed more possession but Ajax attacked with pace. Fraser Forster almost helped the visitors' cause when he spilled a routine cross but the Celtic goalkeeper recovered to kick clear.
Celtic immediately created a glorious chance. Mikael Lustig's low cross should have been knocked home by Teemu Pukki, but the Finland international striker failed to connect with the ball, displaying the kind of profligacy that had been undermining Celtic's Champions League campaign.
Instead of taking the lead Celtic almost went behind on the half hour, after they failed to properly clear a Viktor Fischer free-kick. But when the ball dropped to Christian Poulsen at the back post his effort rebounded off Forster's right-hand post.
At this stage, Ajax were increasingly dominant but when Celtic's breakthrough arrived, it resulted from Denswil's ill-judged challenge on Stokes just inside the penalty area when the forward was not causing any particular trouble. Forrest, hitherto Celtic's best performer, stepped up to stroke home an excellent penalty.
Half-time had been notable for a clash between a section of Ajax fans and police. A steward was hit by a seat apparently thrown from the visiting area, although the fracas was contained swiftly. A stronger than normal security presence had been apparent in Glasgow before the match.
Celtic's second-half task was to protect their lead and Lennon would have stressed that at the interval. Seven minutes after the re-start, Forster once again proved his team's saviour, the England squad player making a fine save when Thulani Serero was sent clean through on goal.
It was rendered even more valuable within a minute when Kayal's 20-yard shot was deflected into the Ajax net off Denswil. Few would have grudged Celtic their first piece of good fortune in this season's competition, with the build-up work of Georgios Samaras worthy of credit.
Ajax responded with Kolbeinn Sigthorsson shooting narrowly wide and Serero, who had provided Ajax's most potent threat, driving over with 12 minutes left.
If that was Celtic's last real moment of panic, they were to end the night with 10 men and having conceded a goal. Biton, who had replaced Lustig, lunged into Serero and was shown a straight red card. "I have no complaints," said Lennon.
Schöne, also a substitute, then curled a shot past Forster from 20 yards. It was a fine goal but irrelevant to the result.