The Dutch champions are bottom of Group H at the halfway stage with a solitary point after James Forrest's 43rd-minute penalty and Beram Kayal's deflected strike early in the second half meant a fine stoppage-time strike from substitute Lasse Schone was no more than scant consolation.
Veteran midfielder Christian Poulsen struck the post with the game goalless after forward Kolbeinn Sigthorsson rippled the side-netting, while Thulani Serero spurned the visitors' two best chances after the break before drawing a foul that saw Celtic substitute Nir Biton sent off.
Ajax might have created even more from lengthy spells of possession, but De Boer was satisfied with their overall performance and insists more of the same will bring the victory he requires when the sides meet again in Amsterdam.
"We had the best chances to score, and they did not," he told UEFA's official website. "But the fact is that in football other laws are counting, so the best team does not always win the match.
"We dominated the game. Poulsen hits the post for us, Serero is two times in an one-on-one situation with their goalkeeper and for Sigthorsson goes the same.
"Those were real goalscoring opportunities for us, so they have to be goals also. But when we play like we have now again in the next match then we certainly will beat them in the Amsterdam ArenA. I really believe in that."
De Boer, who did not dispute the award of Celtic's penalty for Stefano Denswil's clumsy foul on Anthony Stokes, also felt a lack of experience counted against his players amid Celtic Park's cauldron-like atmosphere.
He added on Sky Sports: "Maybe also the youth in my players – they need to have more experience in that kind of situation.
"But I cannot be angry at them. They gave 120 per cent, they created chances but football is about goals.
"Normally the best team wins and I will be satisfied if we play like this and get these chances. In the end, you have to score."
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