Since being appointed in charge of first team affairs in October 2007, Robinson steered Edinburgh to fourth place and second spot finishes in the Magners League while guiding the team to eye-catching Heineken Cup wins over Leinster, Leicester and away at Castres.
After finishing the season as the highest points scoring team in the Magners League, Edinburgh also set a new all-time record for successive victories after winning their final six matches of the season.
As a result, Cartwright expressed his gratitude towards Scotland's new head coach and believes the foundations built during his time associated with Edinburgh will enable the club to challenge for honours in the future.
Cartwright said: "Andy's professionalism and knowledge has enabled Edinburgh to progress at a phenomenal rate over the past two seasons and his contribution, both on and off the pitch, will never be forgotten.
"I would like to thank Andy for his contribution towards elevating Edinburgh into becoming a recognised force in European rugby and his rapport with our players, staff and supporters was incredibly special.
"His top-class coaching skills have enabled our vibrant young team to develop into a team capable of challenging for trophies and now we must carry that positive work forward as we look to progress to the next level.
"On behalf of everyone associated with Edinburgh Rugby Club, I would like to wish him well as he embarks on the next chapter of his coaching career with Scotland."
Robinson leaves behind a squad containing 15 Scotland internationalists, with three players earning their debut caps over the past 18 months, and an average age of just over 24.
Edinburgh finished the previous campaign as the Magners League leading points scorers - notching up 11 more points than champions Munster - and Cartwright believes the future is bright for the team from the Capital.
He said: "We have got a really talented backroom staff and a squad bursting with quality. Andy was always very complimentary about the team that he worked with and that team remains in place. The players have shown what they are capable of and I have every confidence that they will be determined to show that next year will be better still.
"The foundations are now in place for Edinburgh to challenge for trophies and we are excited about what the team can achieve."
On the topic of appointing Robinson's successor, Cartwright says the recruitment process will be thorough and that Edinburgh will leave no stone unturned in locating the most suitable candidate for the job.
He added: "Edinburgh's head coach role is arguably one of the most attractive jobs around and I'm sure we will attract the interest of coaches with the right skill-set and mindset to continue the progress that has been made.
"Whoever is the most suitable applicant will inherit a talented young squad who are eager to learn and with a tremendous team spirit.
"Over the past two seasons, Edinburgh has grown, both in terms of the team and our fanbase, and we are determined to continue that building process until we have reached our objectives; to reach the upper echelons of northern hemisphere club rugby.
"Edinburgh is an ambitious club and, now the first couple of phases have been initiated, we will carry on until we fulfill our potential."
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