The German forward 19th minutes strike was his first for the club since his summer move and owed to his sides positive start to the game.
However, Spurs improved after the restart and deservedly equalised late on when Lamela combined with new recruit Sergio Reguilon to send the game into penalties.
In a match full of drama, Jose Mourinho clashed with former player Frank Lampard in the touchline, although both managers downplayed the incidence in their post match.
Spurs defender, Eric Dier also had to answer a call of nature midway in the second half while his side was still trailingSpurs defender, Eric Dier also had to answer a call of nature midway in the second half while his side was still trailing.
The match went straight to penalties after the ninety minutes with extra time or a replay scrapped before the season even started.
The first nine players from each side duly dispatched their spot kick comfortably, with Mason Mount sending his wide off the post to send Tottenham into the quarter finals draw.