Clarke was left unmarked as he charged in towards the back post to fire Pompey to their third league win in a row.
Bury had the first chance of the match in the 11th minute when Nicky Ajose forced Pompey goalkeeper Luke McGee into a save with a driving shot from just inside the area.
The home side then responded with a chance of their own through Danny Rose after he tried his luck from 40 yards out, only to be denied by Leo Fasan’s fingertips.
Pompey then had another opportunity to go ahead in the 34th minute after a deflected pass reached Gareth Evans, who could only strike his effort at Fasan. Jamal Lowe immediately struck the rebound which went just wide of the post after hitting a defender on its way through.
But it was centre-back Clarke who broke the deadlock in the 62nd minute when he headed in from Evans’ corner.
Manchester United will be watching developments over Gareth Bale, who is set to hold future talks with Real Madrid this summer.
Gareth Bale wants first-team football and will discuss his future with Real Madrid this summer, his agent has confirmed.