Marin Tomasov gave the hosts an early lead with a nicely-taken left-footed effort but the visitors secured a comeback victory with three goals within eight second-half minutes.
Bruno Fernandes started the party in the 48th minutes after slotting from the penalty spot, with winger Gelson Martins and Doumbia scoring the decisive efforts in the 50th and 56th minutes.
Astana's case to salvage a positive result was worsened after Yuri Logvinenko was shown a red card in the 62nd minute after receiving his second booking.
The result has handed Jorge Jesus' side an advantage going into the reverse fixture at the Estádio José Alvalade next week Thursday.
Doumbia has just opened his goal account in the Europa League with his two goals on the continent coming in Sporting's unsuccessful Champions League campaign.