The Dutch champions found themselves 2-0 ahead at half-time thanks to goals from Thulani Serero and Danny Hoesen, but they were forced to play the final 40 minutes without central defender Joel Veltman.
The 21-year-old was dismissed for a challenge on Neymar inside the penalty area, and Xavi halved the deficit with the resulting spot-kick.
Barcelona piled on the pressure but were unable to find an equaliser, and De Boer praised his side's resilience to keep them out and inflict the La Liga powerhouse with their first defeat of the season.
"I don't think Barcelona expected this, we have achieved a victory thanks to our fighting spirit and our quality," he said.
"The first half was fantastic, after that it was unfortunate that we were down to 10 men.
"I had said that Ajax would be the first team to beat Barcelona with the football we always want to play, which is offensive football."
The head coach had sympathy for Veltman after he tried to recover following a poor back pass from Ricardo van Rhijn, but critcised him for not letting goalkeeper Jasper Cillessen deal with the situation.
"It was an expensive lesson for Veltman," he added.
"He should have trusted on his goalkeeper and not have made the foul.
"After his sending off it became difficult and dangerous but we showed our lion’s heart in this match."
Ajax must now win their final game in Group H against Milan to progress to the knockout stages.
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
‘England have the players to win the World Cup – it’ll be tough for Thomas Tuchel to do a bad job, with the squad he has at his disposal’ Former Three Lions winger backs new boss after gentle qualifying draw
