The Black Sticks had got on the board through Blair Hilton's gentle deflection off a hard Alex Shaw drive.
Belgium increased their lead through Boon early in the second half, swivelling away from Shaw to strike the ball past Manchester.
Andy Hayward's low drive from a penalty corner closed the gap again, and had Belgian goalkeeper Vincent Vanasch not managed to turn another Hayward attempt past the post three minutes from the end, the game would have been heading for a penalty shootout.
New Zealand subbed Manchester off to play an extra outfielder for the last five minutes in a desperate attempt to equalise.
"We did really well in the final 15 minutes and almost created a goal right there at the end," New Zealand coach Colin Batch said.
"It's frustrating because we have a feeling that we did enough to get the job done."
World No 2 Australia edged out the Netherlands 4-3 in the other semifinal and will face Belgium in the final.
In other games, Spain beat Ireland 1-0 and India defeated France 6-2, leaving Spain and India to play off for places five and six, with France and Ireland to battle for seventh and eighth.