The launch of the Soyuz was watched by Mr Putin, who had flown 5,500km (3,500 miles) to the country's Far East for the event.
The Vostochny cosmodrome was built to reduce dependency on the Baikonur launch site in Kazakhstan.
The unmanned rocket is carrying three satellites.
It left Earth at 05:01 Moscow time (02:01 GMT).
The initial attempt at launch was called off less than two minutes before lift-off on Wednesday, following a technical fault.
Mr Putin stressed that the delay was related to the rocket itself - not the cosmodrome, located in the Amur region near the Russian-Chinese border.
It was Mr Putin's personal idea to build a new space port in Russia to avoid any potential political risks of using the old Soviet launch centre at Baikonur.
But construction of the vast complex in Russia's Far East has been hit by delays and dogged by corruption scandals.
Four people have already been arrested.
"If their guilt is proven, they will have to change their warm beds at home for plank-beds in prison," the president has warned.
Russia has launched the first rocket from its new Vostochny cosmodrome, following a 24-hour delay that drew the ire of President Vladimir Putin.