Dalziel spent almost two years in charge of the Sevens after being hired in the summer of 2017.
He previously enjoyed a spell as head coach for the Scotland under-20s, while also enjoying success at Premiership Melrose, with whom he won three league titles and a Scottish Cup.
Former Scotland Sevens captain and Rugby Women’s high performance manager Scott Forrest will take temporary charge for the two remaining World Rugby Sevens Series matches.
Dalziel said: “I have really enjoyed my two years working with the sevens team and competing on what is a very competitive circuit. It has greatly helped my personal and professional development and I feel it was successful in developing younger players and fulfilling the remit of enabling them to perform well in sevens and be integrated back into 15s.
“I have always been clear in my ambition to coach at a professional level and my experiences so far at Melrose, London Scottish, age-grades and sevens have all led to that goal.
“I think the hard work starts now, but I am impressed with the set-up Dave Rennie has in place, there is a team approach among the coaches and I’ve worked with many of the Glasgow players through the age-grade system or sevens so I’m very much looking forward to this new stage of my career.”
Scottish Rugby technical director Stephen Gemmell said: “I’d like to thank John for his excellent leadership of the Scotland Sevens team over the past two seasons and wish him all the best in his new role.
“His coaching progression through our club game, his time at London Scottish, age-grade and sevens teams has rightly been rewarded with a role in professional rugby at Glasgow Warriors and follows the footsteps of previous Scotland Sevens coach Calum MacRae into Edinburgh Rugby. This clearly demonstrates the coaching pathway to professional rugby that we have developed within Scotland.”