Firefighters were called out to the ground on Old Dalkeith Road at around 4am where they found two Honda cars and the pavilion on fire.
Police said three men were detained near the scene and inquiries were ongoing.
Firefighters took just over two hours to bring the blaze under control and were today waiting on the electricity supply to be isolated to get in for closer inspection.
A ScottishPower worker said he heard a security guard at Cameron Toll had seen a joyriders on the grass and had called the police.
The council's parks department said the building would be made safe and then demolished.
Mike Shields, parks development officer at the council, in charge of Inch Park, said: "They were changing rooms for cricket and rugby with showers, toilets and changing facilities. I think it was a wooden building, about a few decades old.
"It was used by Lismore Rugby Club and various cricket clubs.
"We'll need to see about getting this made safe and all the utilities turned off.
And we'll look at getting temporary facilities for the club." He added that he did not believe there was any equipment of value in the pavilion and only the remains of a burnt-out cricket roller could be seen today.
A local resident who was walking his dog in the park said the pavilion had long been used by the rugby club and it was used for cricket too.
David Ketchin, 38, a computer programmer, who lives within the grounds of Inch Park, said: "It's appalling.
"Kids break into the pavilion all the time and there are plenty of burnt-out cars round here."
A police spokeswoman said: "We have detained three men for questioning and inquiries are still ongoing. It appears two cars, possibly stolen, were fairly deliberately crashed into the pavilion.
"The cars have been badly damaged. We are appealing for witnesses to come forward."
Councillor Fred Mackintosh, whose Newington ward borders Inch Park, said the pavilion had been set to be pulled down eventually .
He said: "The pavilion was in terrible condition, but was due to be replaced in the long term. The loss of it will be hugely disruptive to local sports users."
A spokeswoman for the fire service said it had sent nearly two dozen firefighters to tackle the blaze.
She said: "Twenty three firefighters went to the scene as well as three pump appliances, which carry water, and three special appliances.
"It was under control by 6.15am."
Lismore Rugby Club have been using the Inch Pavilion for more than 30 years and were due to play a French touring side in Inch Park a week from today.
Written by Evening News; Edinburgh