Cockerill, who has been in charge for four seasons, said he would now “pursue other opportunities”.
Edinburgh endured a difficult season, finishing fifth in Conference B of the Guinness Pro14 and missing out on a place in the Heineken Champions Cup.
It was in stark contrast to the previous campaign where they won their conference and progressed to the Pro14 semi-finals where they suffered a narrow home defeat by Ulster.
The capital side were handicapped this season by being without their Scotland players for the vast majority of the campaign due to the extended international season and Covid restrictions preventing their leaving the Scotland bubble.
Cockerill said the decision had been taken after a meeting with Mark Dodson, Scottish Rugby’s chief executive, and Jim Mallinder, the SRU’s director of rugby.
“Having had constructive conversations with Jim Mallinder and Mark Dodson post season it was agreed that we would come to an agreement where I would leave the club by mutual consent to pursue other opportunities,” said Cockerill.
“I would like to thank everyone at Edinburgh Rugby for the hard work they have put in. It has been a rewarding four seasons for the club.
“I would also like to thank all the supporters for everything they have done for me and the team you have been amazing.
“I wish the staff and players all the best for the future and look forward to watching you in your new stadium."
Edinburgh Rugby Managing Director, Douglas Struth, said: “I would like to thank Richard for all his hard work and commitment to the club throughout his time at Edinburgh Rugby. We wish him all the very best in his next challenge.”
Edinburgh said their pre-season preparations will be led by initially by assistant coach Calum MacRae and head of strength and conditioning Nick Lumley.