Both sides had to make late changes to their starting line-up as Barnet's Simon King was replaced by Anthony Charles and Boston's Stuart Talbot was replaced by Francis Green.
Boston had the first chance of the game as an Alan White header drifted over the bar, but the opening stages were largely uneventful.
Barnet thought they had grabbed the lead on 17 minutes when Ben Strevens turned the ball into the net from close range, but his strike was ruled out for a push.
Soon after Boston carved out another chance following a long ball by Lee Canoville. Striker Laurie Dudfield hit a shot on the turn from 20 yards but his effort flew into the hands of goalkeeper Ross Flitney.
Lee hit a long-range free kick for Barnet straight at keeper Conrad Logan just after the half hour mark, and moments before half-time a cross from Canoville drifted inches over the head of Julian Joachim, who surely would have opened the scoring.
The second period started more brightly, and Barnet grabbed the lead within two minutes as they came out with all guns blazing.
Damian Batt had a shot tipped over immediately from kick off, and on 47 minutes Dean Sinclair delievered a great cross from the right which Lee headed expertly into the top corner.
Five minutes later captain Nicky Bailey came close to doubling the lead, but his 25 yard drive was tipped over by Logan.
Francis Green came closest to an equaliser but his shot flew just wide of Flitney's near post.
Sub Jason Norville almost added a second in injury time but his shot whistled past the post.