The match ended 1-1 after an impressive equaliser in the 88th minute by Dortmund’s Ivan Perisic, but Van Persie has said that the Gunners can be pleased with their display.
He told Sky Sports: “One-one away is not too bad. You come to a difficult place here.
“They have amazing fans and a really good team, so I would have settled for a draw before the game.
“The only thing is that they left it so late, which make it a bit frustrating but we can be happy.”
One of Arsenal’s new signings, Yossi Benayoun, has echoed his captain’s sentiments, and hopes that their recent performances can spur them on in the Premier League.
“It was a good performance away, it’s very difficult in the Champions League away from home," the Israeli said.
“We saw really the two points but they scored at the end and we couldn’t do anything about it so we need to be happy with a point.
“Champions League is always difficult. Dortmund are the German champions.
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“They are a very good team, very difficult. I think we can have a lot of confidence and hopefully continue with it to Blackburn on Saturday.”
Arsenal goalkeeper Wojciech Szczesny, who made a string of stops, echoed Van Persie and Benayoun.
"It’s fair to say it was a defensive performance from us but having said that we handled it really well and we deserved at least a point and I think we’re happy with the performance," he added.
Meanwhile, Arsenal’s assistant manager Pat Rice has spoken about the difficulty of not having Arsene Wenger on the bench, but has said that the team coped admirably without the Frenchman’s direct influence during the game.
Rice said: “It’s strange not being in contact with the boss but we’ve got experience and we did what we thought was right and of course we were playing against a very good side tonight.”
Arsenal face Blackburn Rovers away from home in their next Premier League game and will be looking to maintain their form after beating Swansea 1-0 in the last round of domestic fixtures.