Lester turned in a mishit shot by Peter Till in the 37th minute and although Bournemouth had chances, they could not get past Lee.
Lester had looked certain to put Chesterfield in front in the 15th minute until he was dragged down by keeper Shwan Jalal, who was lucky to escape a red card.
Bournemouth had a good chance to equalise before half-time when Warren Cummings crossed from the left but Liam Feeney sliced his attempted volley.
The visitors pushed hard for an equaliser after the break and were unlucky when Cummings saw a cross bounce back off the bar in the 65th minute.
They had a great chance to level in the 72nd minute when substitute Jamie Lowry handled but Alan Connell's well-struck spot-kick was brilliantly kept out by Lee. The keeper made another great save in the 89th minute when he dived to block a low drive by Mark Molesley.