The Gunners have been all-conquering this season but fell behind to Henrikh Mkhitaryan's 16th-minute opener.
Olivier Giroud levelled before half-time but Dortmund always looked
lively and Lewandowski won it for it with eight minutes left.
The other game in the group also ended 2-1 as Napoli won in Marseille - the French side having lost all three games so far.
Jose Callejon put Napoli ahead three minutes before the break and
Duvan Zapata put Rafael Benitez's side clear as it bounced back from
its weekend loss to Roma.
Andre Ayew struck for the host with four minutes left but it was too
late, with Arsenal, Napoli and Dortmund all tied together on six
Lionel Messi scored on his starting return to Barcelona's line-up as it drew 1-1 with AC Milan at the San Siro.
Messi has been sidelined by injury - which forced him to miss
Argentina's 2014 FIFA World Cup qualifiers last week - but was needed to claim a
point after a defensive howler gifted Milan the lead.
Javier Mascherano and Gerard Pique were at fault to allow Robinho to combine with Kaka before putting his side ahead.
Messi replied on 23 minutes after he was afforded too much space, but
despite the visitor dominating thereafter it was unable to find a
Barcelona tops Group H with seven points, with AC Milan having five. Celtic comes next with three after beating Ajax 2-1.
James Forrest's penalty put it ahead and Beram Kayal made it two.
Nir Biton was then sent off and Lasse Schone netted a consolation.
Chelsea won 3-0 in Germany as it saw off Schalke to go to the top of Group E.
Fernando Torres scored twice for Jose Mourinho's men, with Eden Hazard adding the other on a fine night for the 2012 winner.
The other game in that group saw Basel lose ground as it drew 1-1 with Steaua Bucharest.
Marcelo Diaz put the Swiss side ahead after the interval but with two minutes left Leandro Tatu earned the Romanian outfit a point.
In Group G, Atletico Madrid won 3-0 at Austria Vienna.
Raul Garcia scored once and Diego Costa twice for Diego Simeone's
side, which lost its 100 per cent Primera Division record at the
It heads the group on nine points and is almost on the brink of
qualification, with second-placed Zenit St Petersburg a 1-0 winner at
Porto thanks to Aleksandr Kerzhakov's goal.
Matchday 3 results
Schalke 0-3 Chelsea
Steaua Bucuresti 1-1 FC Basel
Arsenal 1-2 Borussia Dortmund
Marseille 1-2 Napoli
Austria Wien 0-3 Atletico Madrid
FC Porto 0-1 Zenit Petersburg
AC Milan 1-1 Barcelona
Celtic 2-1 Ajax