Starnes will depart Huish Park at the end of this week after almost six years with the Somerset club, culminating in May's promotion to the Championship via the League One play-off final at Wembley.
The 58-year-old, formerly chief exec at Swindon, will also join the board at Plymouth when he starts his new role on Monday.
Argyle chairman James Brent told the club's official website: "My fellow directors and I are delighted to welcome Martyn to Plymouth Argyle.
"Martyn brings years of experience in running football clubs and has demonstrated that, even for a club which has a much smaller fanbase than Argyle, financial prudence can be combined with league success.
"We are confident that Martyn will leverage the additional revenues anticipated from the new grandstand to support John Sheridan in moving Argyle back up the Football League. It is a measure of Martyn's confidence in Argyle that he has decided to join us from a newly promoted Championship club.
"I look forward to the new season with enthusiasm and excitement."
Meanwhile, Yeovil have also wished Starnes well in his new role.
Town's official website read: "All the staff and board of directors would like to put on record their thanks to Mr Starnes for all his work for the club during his time at Huish Park, and wish him the best of luck at Home Park in the future."