The hosts grabbed the lead thanks to Green's superb finish after 16 minutes and the Tigers should now be safe from the drop, while Dario Gradi's side now need a magnificent run to stay in the Championship.
Crewe had a chance earlier on while Steve Jones tried his luck from outside the box, but his well struck shot went just wide of Boaz Myhill's left-hand post.
Hull came back though and took the lead with their first attack. A long ball from Myhill was knocked down by Jon Parkin to Stuart Elliott, who produced a curling cross from the left. The ball fell at the feet of Green who slammed the ball into the back of the net from 12 yards.
Parkin could have doubled the hosts' lead midway through the half as he showed great strength to hold off his marker on the edge of the area, but found keeper Ross Turnbull equal to his powerful strike.
Leon Cort and Craig Fagan also came close before half-time, but Cort's towering header was cleared off the line by Kenny Lunt and Fagan's shot from 12 yards flew over the bar.
Parkin had a chance earlier in the second half as he managed to turn inside the area but his shot went over the bar.
Hull were enjoying most of the play during the second half, but they couldn't take advantage as Crewe were quick to close down and clear.
Both managers were eager for goals and Hull boss Peter Taylor tried to secure a second successive home win by bringing on three attacking players in Billy Paynter, Darryl Duffy and Kevin Ellison.
Dario Gardi threw on his own attacking men in David Vaughan and Michael Higdon in an attempt to salvage something from the game.
Taylor's side nearly doubled his lead in the closing stages as a long throw from Alan Rogers was met by Parkin, who slammed the ball against the crossbar from six yards. Paynter fired the rebound off the bar from the right-hand side of the area.