The 32-year-old struck on five, 11 and from the penalty spot after 35 minutes to hand Sir Bobby Robson’s side another three points, although substitute Marko Babic pulled one back for the visitors, who had seen a 20th-minute Oliver Neuville penalty saved by Shay Given.
As in Germany last week, Newcastle needed just five minutes to force their way in front as Shearer collected his first ever Champions League goal from open play. Robert’s throw-in found Gary Speed in space and the 32-year-old met his far-post cross with a powerful diving header to give keeper Jorg Butt no chance.
It could have been two three minutes later when, after Titus Bramble had robbed Basturk deep inside his own half, Dyer and Shearer combined before the midfielder teed up full-back Andy Griffin to force an excellent save from Butt.
The second goal arrived on 11 minutes when Shearer cut out a Cris’ pass and fed Ameobi, whose low ball was tipped on to the recovering defender by Butt and sat up nicely for the former England captain to head home.
United got a wake-up call with 19 minutes gone when Brazilian striker Franca broke through and went down under Shay Given’s challenge, and although the goalkeeper’s protests proved in vain, he got down superbly to save Neuville’s spot-kick.
Basturk almost got the better of Caldwell two minutes later as the United defence were stretched once again, but Ameobi decided attack was the best form of defence and produced a powerful run and shot which was deflected wide by Diego Placente.
Cris, who suffered an unpleasant evening at the BayArena, was spared further misery when replaced by Zivkovic on 29 minutes, but his arrival coincided with a Newcastle revival.
However, Simak wasted a glorious opportunity to edge his side back into the game 14 minutes before the break when his drag-back defeated both Bernard and Bramble, but he hammered his shot high and wide.
But the respite was short-lived and Shearer completed his hat-trick from the penalty spot with 35 minutes gone after Kleine had hauled back Dyer.
It was his first hat-trick since he smashed five goals past Sheffield Wednesday in an 8-0 win in Robson’s first home game in charge on September 19, 1999.
Ameobi could have made it 4-0 two minutes later but failed to connect with Kerr’s cross in front of goal.
The visitors came close to extending their lead on 55 minutes when Ameobi broke from halfway and despite having Shearer and Dyer in close attendance, shot left-footed just wide from 22 yards.
Griffin had his heart in his mouth on 63 minutes after Simak and Franca made the most of Bramble’s slip to force their way into the penalty area, the full-back diving in to drive the ball just wide of his own goal.
Robson gave Dyer a breather when he replaced him with Nolberto Solano on 70 minutes, but it was Robert who almost put his side 4-0 up with a vicious free-kick from all of 40 yards which was superbly turned away by Butt.
Shearer saw a close-range header from a Robert corner deflected wide seconds later as United stepped up another gear. But Leverkusen finally made their mark on 73 minutes when Babic played a neat one-two with Basturk before lifting a clever shot over Given to reduce the deficit.
NEWCASTLE: Given, Griffin, Caldwell, Bramble, Bernard, Kerr, Dyer, Speed, Robert, Ameobi, Shearer.
BAYER LEVERKUSEN: Butt, Preuss, Kleine, Cris, Placente, Kaluzny, Ramelow, Simak, Basturk, Neuville, Franca.