Fachie, with his pilot Barney Storey, finished with a time of 1:01.351 to guarantee himself a medal with only Kappes left to ride.
On his second attempt the same problem occurred and, as there is a two-start rule in cycling, they were disqualified.
Australia's Kieran Modra set the early pace, clocking 1:03.120 after going out first.
It took seven riders before his time was beaten, when Spaniard Jose Enrique Porto Lareo set a Paralympic record in 1:02.707.
Dutch rider Rinne Oost rode 1:03.52 with his pilot Patrick Bos before Fachie set the world record.