Rovers led 1-0 at half-time courtesy of Malik Wilks' 30th-minute goal, but Bogle's second goal in as many matches since joining on loan from Premier League Cardiff earned the hosts a point.
Pompey are down to third in Sky Bet League One after going four games without a win.
Bogle, who scored on his debut at leaders Luton in midweek, should have scored after only 18 minutes but put a free header straight at goalkeeper Marko Marosi.
And the visitors took the lead when early Pompey substitute Christian Burgess was caught under a long punt downfield and Wilk raced clear to slot under the hesitant Craig MacGillivray.
The home side equalised early in the second half following a fumble by Marosi. Substitute James Vaughan collected the loose ball and rounded the stricken keeper before squaring for Bogle to net from close range.
Doncaster striker John Marquis forced MacGillivray to tip a curling effort round the post at full stretch in the 73rd minute.
And the visitors nearly won it late on but Alfie May could not beat MacGillivray.