The group said they were campaigning against Trinity College's plans to sell off land they own in Suffolk to someone with their sights set on developing a lorry park for thousands of vehicles.
"We must call time on those profiting from the destruction of nature," XR Cambridge said.
Images showed tufts of grass scattered over the lawn in the aftermath of the protest.
A college spokesperson said: “Trinity College regrets the criminal damage done to its property beside Great Gate.
"The College respects the right to freedom of speech and non-violent protest but draws the line at criminal damage and asked the protestors to leave."
Cambridge Police said they were working with the college following the incident.
"A crime has been recorded for criminal damage," they said.
Trinity College owns Innocence Farm in Suffolk and is attempting to sell the land to the Port of Felixstowe so they can develop a lorry park for 3,000 vehicles.
We must call time on those profiting from the destruction of nature. https://t.co/ZCqKoBVp6B— XR Cambridge (@xr_cambridge)February 17, 2020
XR Cambridge blocked a road in the city on Sunday to campaign for greater action against climate change, the group said on social media.
They said activists plan to cut off Trumpington Road in a week-long protest "to force local institutions to take action on the climate emergency".
Following the demonstration on Trinity College's lawn, a spokesperson said: "Academics at Trinity are actively engaged in research to understand and develop solutions to climate change, and taking practical steps forward.
"The College fully supports the University initiative, Cambridge Zero.”
Extinction Rebellion also demonstrated outside the International Maritime Organisation's headquarters in London on Monday.
The protest, which saw activists dressed as polar bears, aimed to call on the UN's international shipping regulator to introduce stricter rules on exhaust pollution from ships.
Climate activists also targeted London Fashion Week last week, calling for the event to be cancelled over concerns about the clothing industry's environmental impact.
Additional reporting by Press Association.