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.
Lewandowski, who had himself earlier squandered a close-range effort, made amends with three minutes left when he controlled the ball in the box, turned and fired past goalkeeper Kenneth Vermeer.
"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."
TOP TWO
With heavyweights Manchester City and Real Madrid also in the group and Real winning their first encounter, Dortmund are hoping to crash the party and finish in the top two to advance to the knockout stage.
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.
Ajax survived a couple of early scares when the hosts pressed high, winning possession but failing to finish their quick breaks.
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.
Babel, resuming his relationship with Ajax after spells at Liverpool and Hoffenheim, continued being a threat but Dortmund gradually upped the pace and threatened with shots from Lewandowski and Reus.
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.
His blushes were later spared by his striker, who was left unmarked for too long in the box to drill in the winner.
"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."
Ajax next host Real on October 3 when Dortmund visit City.
