The visitors enjoyed the majority of chances and deservedly took a 16th-minute lead thanks to Bryan Ruiz's strike. However, they were punished for failing to take their opportunities when Gokhan Tore equalised for the Turkish side after 61 minutes.
Besiktas were on a high having beaten Skenderbeu 1-0 in Albania in their opening group game and also defeated league rivals Fenerbahce 3-2 in the Istanbul derby on Sunday. But Sporting, who opened their Europa League campaign with a shock 3-1 home defeat to Lokomotiv Moscow, started the better.
The visitors had a superb chance to take the lead after 12 minutes. Teofilo Gutierrez was picked out with a long ball forward but his effort from 10 yards was well saved by Besiktas goalkeeper Tolga Zengin.
Four minutes later, however, Sporting made the breakthrough. Matheus Pereira's cross from the right fell to former Fulham player Ruiz on the edge of the penalty area, and his left-footed strike found the bottom corner beyond Zengin.
The Portuguese side had made a promising start and continued to press their hosts.
With half-time approaching, the lively Gutierrez should have made it 2-0. He raced on to a neat ball over the top, and with space in Besiktas' penalty area he flashed his shot narrowly wide from an angle.
Sporting continued to press after the interval and Gutierrez was at the heart of the action again. He was played through by Carlos Mane and although his shot was well struck, it was straight at Zengin.
The visitors were punished for their misses when Besiktas equalised. Jose Sosa's clever back-heel pass on the edge of the area set up Tore, who calmly slotted beyond Sporting goalkeeper Rui Patricio.
Besiktas had their tails up and Cenk Tosun almost made it 2-1 but his shot flew wide of the left post.