The 41-year-old, appointed on a three-year deal in May, becomes the first managerial departure of the new season.
Saturday's 2-1 defeat at Southend left them 20th, with three wins from their last 17 games in all competitions.
"I felt it was time to take action and make a change," Posh chairman Darragh MacAnthony said in a club statement.
"It had become plain to see our young squad was under-performing and playing nowhere near their potential.
"Make no mistake, our players must also shoulder responsibility for this awful start to the season which has now led to a great person losing their job."
Robertson finished last season in caretaker charge of the Cambridgeshire club, taking over after Darren Ferguson left the ABAX Stadium in February.
Peterborough's sole league win this season came against Colchester last month, and the Southend defeat left them just above the relegation zone.
"Thanks for the journey and I genuinely wish all at the club the very best of luck for the future," Robertson wrote on Twitter.
MacAnthony added in his statement that he was "driven every day" to improve the club, with the aim of returning to the Championship.
"It has been said by some fans this season that I have lost my own hunger and appetite as chairman, that I haven't made the necessary investment to take the team forward," he said.
"Nothing could be further from the truth.
"I will begin the recruitment process of finding a new first team manager who will be tasked with putting our promotion aspirations back on track this season and quickly."
League One side Peterborough United have sacked manager Dave Robertson after winning just one of their first six league games this season.