Lewandowski held off Per Mertesacker in his own half to instigate the breakaway he finished from Kevin Großkreutz's cross for a classic Dortmund goal.
Henrikh Mkhitaryan put last season's European Cup runners-up in front but Olivier Giroud equalised just before half-time, thanks to some hesitant goalkeeping from Roman Weidenfeller.
Dortmund ran four kilometres more than Arsenal in the first 45 and the Gunners were the more dominant side after the pause. Chances were at a minimum, but substitute Santi Cazorla hit the bar.
However Lewandowski, perhaps fortunate not to have been sent off for an elbow on Laurent Koscielny, put Dortmund level on points with Arsenal in Group F.
In one of the tightest groups, Borussia, Arsenal and Napoli are all on six points with the risible Marseille pointless.
Lewandowski celebrates his winning goal
Arsenal travel to Dortmund in a fortnight, with the daunting task of becoming the first European opponents to win at Signal Iduna Park in nearly two years.