Murray left his post as a first-team coach at Oldham Athletic to take charge of his former club this week, but there was no dream start for the 38-year-old.
Brad Walker cancelled out Ryan Donaldson's 26th-minute opener with just under an hour played and that looked to have secured a commendable point for struggling Hartlepool.
However, Harrison Dunk netted the winner three minutes from time to ensure the visitors slipped to the bottom of the table.
There has also been a change at the top of the table, as a seventh successive league win for Luton Town propelled them above Wycome Wanderers.
Northampton Town were their visitors on Saturday and they looked destined to leave a packed Kenilworth Road with a point, but Luke Gutteridge's 90th-minute goal against his former club saw Luton claim a late 1-0 triumph.
Wycombe were surprisingly held to a 1-1 draw at home by struggling Dagenham and Redbridge, with Peter Murphy's 83rd-minute strike ensuring Gareth Ainsworth's men salvaged a point.
Burton Albion were the other side to be overtaken by Luton, with an Ian Sharps own-goal six minutes from the end doing the damage in a 1-0 loss at Stevenage, while Shrewsbury Town's 2-1 win over Portsmouth moved them up to fourth.
The biggest victory of the day saw Plymouth Argyle beat Cheltenham Town 3-0 to solidify their place in the top seven, with the only other win coming in the shape of Carlisle United's 2-1 success over Oxford United.
An Armand Gnanduillet double rescued a 2-2 draw for Tranmere Rovers at AFC Wimbledon to ensure they moved off the bottom, while Mansfield Town were held to a fourth consecutive stalemate as they drew 1-1 at York City.
Southend United and Newport County were also held to 1-1 draws at home, with Bury and Accrington Stanley their respective opponents.