The West Ham goalkeeper inexplicably spilt Tamir Cohen's innocuous header and Ivan Klasnic pounced to score the game's decisive goal with 13 minutes left.
Gary Cahill made certain of the points with a fine header late on to seal Wanderers' first win in eight attempts and heap the pressure on Hammers boss Gianfranco Zola.
Earlier Allesandro Diamanti had given West Ham hope when he cancelled out Chung-Yong Lee's 64th-minute opener.
Bolton dominated most of the first period and were nearly ahead after just 34 seconds when Kevin Davies' deflected shot flew inches wide.
The Hammers' only real chance of the half fell to Kieron Dyer who shot tamely at Jussi Jaaskelainen before limping off soon after with yet another injury.
Wanderers felt they should have had a penalty on the half-hour mark when Julien Faubert clearly handled inside the area, but the referee waved away the protests.
Megson's men continued to pile forward, creating a succession of chances the best of which fell to Zat Knight but the giant defender headed Lee's corner over the bar on 36 minutes.
Both sides came flying out for the second half as the game came to life. Herita Illunga scrambled Cohen's effort clear before Knight made a superb challenge to deny Diamanti.
Green then saved well after Cahill let fly and, seconds later, Jaaskelainen produced a wonderful save to tip over Scott Parker's piledriver on 55 minutes.
Finally the opening goal came when Klasnic slipped in an offside-looking Lee and he lifted the ball over the onrushing Green.
But the Hammers were level just five minutes later when Jack Collison teed-up Diamanti and the unmarked Italian rifled home.
But Green's blunder gifted Bolton the points as Klasnic struck on 77 minutes - and Cahill headed home Ricardo Gardner's corner with two minutes left.