For the home side, a 20th game without a win equalled an unwanted club record set in 1993.
Pools started brightly and Boro goalkeeper Steve Arnold saved from Jordan Richards and Charlie Wyke in quick succession, but the visitors scored with their first attack.
Luke Freeman's cross was met by Marcus Haber and, after his header cracked back off the post, he reacted quickest to prod in the rebound.
Stevenage made the game safe after 60 minutes from the penalty spot. Jordan Richards handled a deep cross and Lucas Akins rolled the ball in.