With the game heading for penalties, Langerman took a speculative shot from 30 yards out which completely deceived the Dutchman and sailed into the back of the net.
In an entertaining game at the Cape Town Stadium, Ajax had the first chance on 22 minutes when Musa Lebusa was left unmarked from a free-kick, but he headed straight at Wayne Sandilands.
The home side kept up the pressure and had another glorious chance four minutes later when Lebohang Manyama fired over from close range after good work by Wanderson.
Sundowns got their first real opportunity on the stroke of half-time when he found himself one-on-one with Westerveld, but he fired straight at the Dutch goalie.
Paul Rusike was the next to find himself with just the keeper to beat, but he too failed to make the most of it as he shot wide.
Downs were much better in the second half, and should have scored the winner four minutes from time when Matthew Booth was dispossessed, but the resulting cross was headed over by the unmarked Katlego Mphela.
Extra-time brought more chances, with the best falling to Nyasha Mushekwi, whose looping header beat the keeper, but came back off the woodwork and was cleared to safety.
Then, with three minutes to go, came Westerveld’s mistake when he completely misjudged Langerman’s shot to see Ajax crashing out of the Cup.
Ajax Cape Town: Westerveld, Lolo, Allie, Booth, Lebusa, Graham, Wanderson (Scott 84), Shabalala, Manyama (Khenyeza 66), Davids, Rusike (Billiat 46)
Mamelodi Sundowns: Sandilands, Mwanjale, Schut (Arendse 62), Mabaso, Langerman, Kekana, Nyandoro, Pelembe, Laffor, Mphela (Manqele 87), Modise (Mushekwi 52)