Rush, alongside brakeman Jesse Lumsden, led from start to finish in Germany as only one home sled finished in the top three that of Francesco Friedrich and partner Gino Gerhardi in third.
Latvia's Oskars Melbardis claimed second while former British sprinter Pickering, pushing for the first time at a World Cup event, and pilot Jackson finished joint 19th.
Rush was the only pilot to break the 50-second barrier during the first run, clocking 49.98 seconds, however Melbardis wasn't too far away and just 0.05 behind in second.
However Rush wasn't troubled by the Latvian and his second run of 50.24 was far better than Melbardis' with the Canadian's overall winning margin 0.38.
Friedrich moved up from fourth to second ahead of Alexandr Zubkov while Jackson and Pickering clocked 50.74 on their first run and 51.17 on their second.
Jackson and Pickering were joined in Koenigssee by another British crew but Lamin Deen and Ben Simons only took to the ice once after their effort of 51.48 put them outside the top 20 in 29th.