Robert Lewandowksi's 87th minute goal secured Borussia Dortmund's first Champions League win of the campaign after what captain Sebastian Kehl described as a "very intense" game.
Dortmund had been under pressure going in to the match, with much having been written about their disappointing Champions League campaign last season and this season's "group of death".
The game against Ajax, in a group that also contains Manchester City and Real Madrid, had been described by many pundits as a must-win encounter - and Lewandowski obliged late on.
"It was a very intense game, up and down," Kehl said. "In the second half, we were much better and we were rewarded for our good work in the 87th minute."
"In such a group, heroes are born," Dortmund chief executive Aki Watzke said after the draw had been made - and both Lewandowski and keeper Roman Weidenfeller were in that category in Dortmund on Tuesday night.
Weidenfeller saved the German champions from falling behind on numerous occasions before the home team turned things around. "It was very important to leave the pitch with positive feelings", he said afterwards.
Borussia started the game unusually nervously, but things began to turn around when Lewandowski fired narrowly over the bar early in the second half.
Mats Hummels then saw a weak penalty saved, and said afterwards: "It would have hurt me if we had lost two points because of that. It was a huge opportunity to take the lead and such a lousy shot."
But Poland international Lewandowski was there to win Dortmund the points late on, and coach Jurgen Klopp said: "It was an impressive goal by Lewandowski. He stayed calm and put the ball away.
"The team's progression has been going on for a while, we continue to mature. There were some things we could have done much better, but we are pleased."