Veteran striker Lee, who cleaned new Town boss Simon Grayson's boots in their playing days at Aston Villa, fired in from Joey Gudjonsson's pass in the 75th minute.
Lee had entered the contest 13 minutes earlier and sparked the game into life after an hour devoid of action and when the Terriers did get out of first gear, they found Pools goalkeeper Scott Flinders in solid form. He first tipped over Antony Kay's flicked effort and then saved Danny Ward's shot.
Flinders also did well to beat away a goalbound header from Jordan Rhodes early on, while at the other end it took until 67 minute before the visitors mustered a shot at goal.
But Hartlepool did rally late on and nearly levelled when Antony Sweeney saw his shot tipped on to the post by Huddersfield goalkeeper Alex Smithies.
Academy product Smithies was also alert in the last minute to make a double stop, first from Sweeney and then Adam Boyd, which was ruled offside.