With the party topping opinion polls in the run up to elections later this year, protesters chanted "Yes we can!" as they made their way from Madrid city hall to the central Puerta del Sol square.
Many waved blue and white Greek flags and red and white Syriza flags or held signs reading "The change is now" and "Together we can".
Podemos, which means "We Can", was formed just a year ago, but produced a major shock by winning five seats in elections for the European Parliament in May.
"We dream but we take our dream seriously. More has been done in Greece in six days than many governments did in years."
Syriza beat mainstream Greek parties by pledging to end austerity and corruption, as Podemos aims to do in Spain's general election due in November.
Iglesias, a 36-year-old pony-tailed former university professor, appeared alongside Syriza's Alexis Tsipras, now Greece's prime minister, to publicly support him during his campaign.
The party has struck a chord with Spaniards enraged by a string of corruption scandals, as well as public spending cuts imposed by the conservative ruling party and previously by the Socialists after the economic crisis erupted in 2008.
"There are many people that agree with the need for change. Enough already with stealing -- that the corrupt take everything and we can't do anything," said Dori Sanchez, 23, an unemployed teacher who came from Monover in southeastern Spain for the rally.
"I want real change, that they stop fooling us," said Blanca Salazar, 53, a geriatric aide who came by car from the northern city of Bilbao with her husband and nephews.
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