Jurgen Klopp’s men had trailed 3-1 with under 30 minutes to go, then led 4-3 before Sebastien Bassong’s stoppage-time strike levelled matters.
But there was still time for substitute Lallana to mis-hit a shot into the ground and secure a 5-4 victory as Liverpool’s own defensive failings were eclipsed by Norwich’s at Carrow Road.
Klopp celebrated on the pitch with a shirtless Lallana, but there is much work to do if the Reds are to advance to the Capital One Cup final at the expense of Stoke, visitors to Anfield for the semi-final second leg on Tuesday night.
It had looked like new signing Steven Naismith would be an instant hero for the Canaries after Dieumerci Mbokani cancelled out Roberto Firmino’s opener with a brilliant back-heel.
Naismith scored his first Everton goal against Liverpool and, on his debut following his move from Goodison Park, the Scot struck again to give the Canaries a 2-1 half-time lead.
Poor defending allowed Norwich to further extend their advantage when Alberto Moreno felled Naismith in the box and Wes Hoolahan converted the resulting penalty.
Jordan Henderson reduced the deficit from a gaping hole in the Norwich defence and another appeared allowing Firmino to equalise with 27 minutes remaining.
James Milner converted for 4-3 after capitalising on Russell Martin’s dreadful pass back before the late drama.
Milner intercepted and finished calmly to restore Liverpool’s advantage only for Bassong to finish superbly from 22 yards for what he thought was the equaliser.
There was another twist to come, though, as Norwich failed to clear and Lallana’s shot looped over Rudd.
Know more about Sport360 Application