Fenerbahce, who wasted a first-half spot-kick, also gained an advantage by beating Portuguese club Benfica 1-0 in Istanbul with a 72nd-minute header from Egemen Korkmaz.
Chelsea, Champions League winners for the first time in 2012, made a tame defence of their continental trophy earlier in the season, exiting in the group stage, but have found the Europa League more to their liking.
The Premier League side appeared to have been denied victory by Basel, cheered on by Swiss tennis great Roger Federer in his hometown, when Fabian Schaer struck an 87th-minute penalty past Petr Cech to cancel out Victor Moses's 12th-minute header.
But four minutes into stoppage time Luiz struck a low freekick that keeper Yann Sommer could only help into the net.
Both semi-finals, while lacking the spectacular fireworks that lit up the Champions League earlier this week when Bayern Munich and Borussia Dortmund each struck four goals past Barcelona and Real Madrid respectively, produced entertaining encounters.
Basel, who knocked out Premier League side Tottenham Hotspur in the quarter-finals, struck the post early in the second-half, as did Chelsea striker Fernando Torres a few minutes later.
The woodwork was also rattled a number of times in Istanbul.
Fenerbahce's Brazilian midfielder Cristian drilled a penalty against the left hand upright on the stroke of halftime after Ola John, who was booked and will miss the return leg, clipped the legs of Gokhan Gonul.
Former Liverpool striker Dirk Kuyt fired against the post after the break, as did Benfica's Nicolas Gaitan before Korkmaz headed in from a corner for the game's only goal - the ball bouncing in off the underside of the crossbar.
The second legs take place on May 2 and the final will be played in Amsterdam on May 15.
