Peter Hartley and Gary Liddle netted the goals as Pools saw off a visiting side who were down to 10 men for the closing stages following the dismissal of Lee Tomlin.
In an game which never let up from start to finish, Andy Monkhouse hit a post after just nine minutes with a clever stooping header which beat Joe Lewis but not the frame of the goal.
But with 44 minutes on the clock Pools took the lead when Evan Horwood's inswinging corner was powerfully headed home from eight yards by Peter Hartley.
Posh were again denied by Kean after 72 minutes, Tomlin's 25-yard free-kick heading for the top corner before the keeper stretched to push it over.
Tomlin then saw red with eight minutes remaining for a lunge on Leon McSweeney before Posh almost levelled when Boyd's curling shot was clawed out by Kean, but there was time for Hartlepool to net a second, Liddle sweeping into the bottom corner in the 89th minute from distance to secure three successive wins.