Niall Ennis pulled a goal back for Argyle, who piled on the pressure in the second half in search of an equaliser.
Doncaster took the lead after seven minutes when Anderson stooped to head in a corner from James Coppinger, who was making his 600th league appearance for the club.
And they extended their advantage after 47 minutes when Bogle headed in a Coppinger cross from close range.
But the cushion did not last long as Plymouth pulled a goal back two minutes later when Ennis, who spent last season on loan at Doncaster, touched in Conor Grant’s cross.
The goal buoyed Plymouth and they were unfortunate not to at least equalise during a strong spell.
Joe Edwards missed a close range header, while vital blocks from Anderson and Joe Wright denied Danny Mayor and Grant.
Ennis missed an excellent opportunity from a header as Plymouth continued to push but Rovers saw the game out.
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