Lester scored his 15th goal of the season in the first half and Lee saved a penalty for the second home game in a row to make sure his team denied Bournemouth a share of the points.
The visitors certainly fought hard and Chesterfield had to dig deep after the break when they were put under pressure for long spells.
It was a different story in the first 45 minutes and Chesterfield might have had a more comfortable evening if referee Mark Haywood had red-carded Bournemouth goalkeeper Shwan Jalal after only 15 minutes.
The keeper dragged down Lester outside his area when the striker had gone past him but he escaped with a yellow although Lester beat him just before the interval.
A long throw was volleyed goalwards by Peter Till and Lester reacted smartly to stab a shot past Jalal from just outside the six-yard box.
Chesterfield could have increased their lead early in the second half but Bournemouth became more threatening when Alan Connell replaced Liam Feeney in the 53rd minute.
Connell saw a right-foot shot go just wide four minutes later and Warren Cummings was unlucky when his 65th-minute cross bounced off the crossbar.
Chesterfield were defending desperately at times but, in the 72nd minute, Jamie Lowry was adjudged to have handled when the ball struck him on the forearm but Lee threw himself to his right to claw out Connell's fiercely-hit penalty.
Bournemouth refused to accept defeat and they almost drew level in the 89th minute when Mark Molesley drilled a low shot through a crowded area only for Lee to plunge to his right and make another excellent save.
Chesterfield had to hang on through five minutes of stoppage time and Alan Goodall almost scored a spectacular goal from inside his own half as Bournemouth threw everyone forward.