Cambridge-based student James Matthews, 25, was also ordered to pay £250 in compensation to the Royal Parks Agency for the damage he caused.
He pleaded guilty at Horseferry Road Magistrates' Court to intentionally or recklessly damaging the statue in Parliament Square.
His defacement of the statue happened on the same day as rioting on 1 May, which saw damage estimated at thousands of pounds caused to central London property.
There was widespread outcry from MPs and the press after the statue of the former prime minister was defaced with red paint and the Cenotaph was sprayed with graffiti during rioting at the anti-capitalism demonstration.
The figure, which stands in Parliament Square, was made to look as though blood was dripping from its mouth.
Graffiti was sprayed on the plinth and a turf mohican was added to the statue's head.
Mr Matthews, who served with the Royal Marines in Bosnia and Croatia, justified his actions by telling court: "I thought that on a day when people all over the world were gathering to express their human rights and the right to freedom of speech, I would express a challenge to an icon of the British establishment."
Magistrate Roger Davies, sentencing him, described the defendant as an intelligent man who should have acted in a "more sober and sensible way".
He added: "Your actions caused great affront to many British people and people overseas.
"The Churchill statue symbolises to many people the war effort and the struggle against the Nazis."
Mr Matthews, who is studying English and European literature at Anglia Polytechnic University, turned himself in to police after a newspaper photograph pictured him hanging from the Churchill statue.
Although he admitted defacing that statue, he denied any involvement in graffiti sprayed on the Whitehall Cenotaph during the May Day demonstrations.
The ex-soldier said it was "a monument to ordinary soldiers and I was an ordinary soldier".