The German champions, looking to make an impact in Group D and in Europe after winning consecutive Bundesliga titles, were frustrated by a tough Ajax defence for much of the game and also saw Mats Hummels' 58th minute spot-kick saved.
Eager to forget last term's woeful Champions League campaign, where they finished last in their group, Dortmund kept pushing late in the game and were rewarded with a master class in clinical finishing - something they had been lacking for most of the game.
"Starting with three points is something that gives us confidence," said forward Marco Reus. "But we have to keep improving and there a lot of matches still to come."
With last season's group haul only amounting to four points, 1997 winners Dortmund will now hope to improve on that.
"It was an unbelievable game, a very intense one," said Dortmund coach Jurgen Klopp. "We defended well over 90 minutes and constantly created chances. It was a challenge to stay patient.
"The team has been developing well for some time now and we are continuing to do so. I am satisfied."
With the BVB stadium's steep stands draped in a sea of yellow-clad fans and with in-form Reus slipping back into the starting line-up, Dortmund shot out of the blocks.
It was, however, the Dutch champions who had the first half's biggest chance in the 12th minute with Ryan Babel racing clear and setting up Christian Eriksen, whose effort was denied by keeper Roman Weidenfeller.
They were rewarded for their offensive play when Mario Gotze was brought down by Ricardo van Rhijn with a clumsy challenge but Germany defender Hummels' weak, low spot-kick was easy work for Vermeer.
"I told the boys they can be proud of the game they played," Ajax coach Frank de Boer told reporters.
"We need to be more cold-blooded. We failed to take the lead. We cannot have chances like that and not take them. Not against teams like Dortmund."
