But the former world No1 amateur made six bogeys and a birdie in his final 10 holes.
Lee was ranked 116th on the FedEx Cup standings going into The Barclays but jumped to 96th after his latest result.
That means he will qualify for the Deutsche Bank Championship this weekend, the second of four tournaments in the FedEx Cup playoffs.
As has been the case since the inception of the FedEx Cup, the ranking system for the PGA Tour, the tournament fields decrease in size as the playoffs continue.
The playoffs began with The Barclays, which was reserved for the top 125 players.
The top 100 then advance to the Deutsche Bank Championship.