A judge ruled that at two different locations where accidents occurred to Elizabeth Fairley and Iain Lowdean the infrastructure, made up of the road layout and tram tracks, posed "a relevant hazard" to the cyclists.
Lady Wolffe said: "There was no breach of duty on the part of either pursuer; they bore no responsibility in law for the accidents that befell them."
Both accident victims were injured while cycling across tram tracks in the city: Ms Fairley at Haymarket and Mr Lowdean at Princes Street.
She raised an action against Edinburgh Trams Ltd and the city council and Mr Lowdean sued Transport Initiatives Edinburgh and the council. Damages in the claims were agreed but liability was contested.
Lady Wolffe said: "There have been numerous other accidents involving cyclists and the tram infrastructure. These two actions are the first of these claims to come to proof."
Nurse Ms Fairley earlier told the Court of Session in Edinburgh that she was thrown from her bike as she negotiated tram tracks in the city.
She said she had set out to cycle to her home in the Corstorphine area of the city from her work at the Sick Children's Hospital in Edinburgh when the accident occurred on October 16 in 2013.
The advanced nurse practitioner said she travelled down Morrison Street to Haymarket before she was injured on the semi-dark, wet evening.
Ms Fairley, 58, said: "I crossed there to get across both tram tracks, but I had to straighten up because there were cars."
She said she knew her front wheel got over and her intention was to get both over before straightening up and continuing her journey westwards.
She said: "That was the aim to get right over because I know tram tracks pull bikes right into them."
"I was looking at the front wheel and trying to get that over and trying to avoid the cars passing. Something pulled me into the tram track and threw me over in the path of the cars that were overtaking me," she told the court
"I have to think it was the back wheel slipping back into the tram tracks. The rear wheel, I think, is the one that got caught in the tram tracks," she said.
"It all happened in a split second. The bike got thrown over. I got thrown over to the right-hand side and fell on the road," she said.
The nurse, who was wearing a cycling helmet, said: "I knew I was injured, everything felt sore." She said she suffered injuries to a knee and her chin and a black eye appeared the following day.
She said a previous incident had made her aware of the danger of a wheel getting caught in the tracks.
"I think I knew from that previous experience you had to cross them, if at all possible, at 90 degrees. It is not always possible, but anything to avoid your wheel getting dragged back into the tram tracks," she told the court.
The injured cyclist raised an action against Edinburgh Trams and the city's local authority, who are contesting liability in the case.
She originally sued for £50,000, but the court heard the amount of damages to be paid if her claim succeeded was agreed. A similar agreement was reached in the second case involving Mr Lowdean
Mr Lowdean, 35, from Edinburgh, originally sued for £15,000, after he fell when his bike slipped on the tram tracks in Princes Street in October 2012. He suffered injuries to his hands and right knee.
Other claims brought by cyclists were put on hold while the lead cases proceed.
It was claimed in the action: "It has been well known for many years that the presence of tram tracks in an urban street poses a danger to cyclists. This was apparent when trams were part of the public transport infrastructure in the 1950s."
It is said that the reintroduction of trams led to a greater appreciation of the danger posed by tracks and the measures necessary to minimise it.
A report produced in the 1990s highlighted the hazard of cycle wheels being trapped or skidding on the flange groove of a tram track and noted the need for cyclists to cross them at as perpendicular an angle as possible.
The council had maintained that it fulfilled all duties of reasonable care.