Former Posh striker Constantine – he cost Torquay £75,000 just before Christmas – missed enough chances to win the relegation six-pointer on his own, but Akinfenwa did the business for him.
After a bright Peterborough start, Torquay took the lead in the 24th minute when Akinfenwa turned smartly in a crowded goalmouth and fired his ninth goal of the season from eight yards.
Torquay had earlier had a Craig Taylor header disallowed because Alex Russell's corner had floated over the by-line, but Akinfenwa's goal boosted their confidence.
Constantine missed two one-on-one chances before half-time, Posh keeper Mark Tyler foiling him each time.
It was a similar story after the break, Constantine missing from point-blank range after a brilliant run and cross by Jo Kuffour, and then pulling a flying save from Tyler after a mistake by Simon Rea.
Posh finally punished those misses when Calum Willock volleyed his 11th goal of the season from a deflected clearance in the 63rd minute.
But Akinfenwa was in no mood to let matters rest there, and in the 74th minute he held off two challenges in the area before curling a 12-yard shot round Tyler for the winner.
The result lifted Torquay three points clear of the League One drop-zone and pushed Peterborough deeper into trouble.