"The police really have a hard time because they're trying to separate these two groups - the neo-Nazis, which want to march and the left-wing people."
Scuffles also broke out between police and protesters from both sides, with several barricades set on fire and some minor injuries reported.
The February 13 anniversary of the bombing of Dresden has become a focus for neo-Nazi groups in the past few years. Some far-right politicians have also sparked controversy by describing the raids as a "bombing Holocaust".
But Helma Orosz, the mayor of Dresden, said the city does not want the neo-Nazis there, the Associated Press news agency reported.
"This gang doesn't belong here"
Helma Orosz, Dresden mayor
"This gang doesn't belong here," she said, following a legal attempt to stop the neo-Nazi march failed.
The human chain helped "make Dresden a fortress against intolerance and stupidity, and the anniversary provided a reminder of who started the war", Orosz said.
The Dresden air raids are considered one of the most controversial attacks by Allied forces during World War II.
More than 30sq km of the historic city was destroyed in heavy bombing by US and British forces.
The number of civilian casualties is disputed, with estimates ranging from 25,000 to more than 100,000 people killed in the raids.