The Swiss champions, who drew 2-2 with Spurs in London last week, again produced a 2-2 scoreline before hammering the visitors 4-1 in a shootout after extra-time.
Basel then hit the front on the night and outright in the two-legged tie when Dragovic was on hand to slot home after Friedel could only partially parry Fabian Schar’s flick-on.
A brave Tottenham fought back yet again, after coming from two goals down last week, with Dempsey rifling home substitute Tom Huddlestone’s clever chipped ball in the 82nd minute.
That made the score 4-4 on aggregate and tied on away goals forcing the match into extra-time but not before Spurs were reduced to ten men when defender Jan Vertonghen was sent off for a professional foul as the last man on Streller.
Tottenham were made to battle out the 30 minutes of extra-time a man down and although they got that far, it ended in the spot-kick competition with Huddlestone having his effort saved and Emmanuel Adebayor completely miscuing as Basel converted four times through Schar, Streller, Fabian Frei and Marcelo Diaz to move into the final four.
2012/13 UEFA Europa League quarter-final – second leg
St Jakob-Park, Basel – 40,000
Red card: TOT - Jan Vertonghen 90+1’