"We hope that the country's new administration collaborates in this sense in order to reach solutions on international and regional questions," Westerwelle said in a statement.
Rowhani, 64, was declared Iran's new president, ending eight years of conservative grip on the top office in a victory cheered in the streets of Tehran and cautiously welcomed by world powers.
Iran's contentious nuclear programme has caused a standoff with the West, which suspects it is aimed at building nuclear weapons something Tehran has repeatedly denied.
In his first statement after his win was confirmed, he called on world powers to treat Iran with respect and recognise its rights, an apparent allusion to its controversial nuclear policy.
"This is a victory of intelligence, of moderation, of progress... over extremism," Rowhani said.
"A new opportunity has been created by this great epic, and the nations who tout democracy and open dialogue should speak to the Iranian people with respect and recognise the rights of the Islamic republic," he said.
Then they will "hear an appropriate response," he added.
Rowhani is a former top nuclear negotiator who has championed more constructive engagement with world powers. He won outright against five conservative candidates with 18.6 million votes, Interior Minister Mohammad Mostafa Najjar said.
That was enough to ensure there would be no run-off against the runner-up,
Tehran mayor Mohammad Baqer Qalibaf, who came a distant second with 6.07
Najjar said 36.7 million people, or 72.7 percent of the electorate, had voted.
Outgoing President Mahmoud Ahmadinejad was constitutionally barred from standing again, having served two consecutive terms.