The Portuguese club trailed to a Marin Tomasov goal after just seven minutes and took time to get back into the contest at the Astana Arena.
Astana, who played the final half-hour with 10 men following Yuri Logvinenko's red card, found themselves in front early on through a smart left-footed finish from Tomasov.
The visitors tried desperately to find a response before the interval, coming close to doing so through a couple of Bryan Ruiz shots that Nenad Eric was equal to on both occasions - the second being pushed over the bar.
Sporting did have a goal ruled out for offside after Doumbia found a way through, but Patricio was again required prior to that with a double stop as the home side took a slender lead into the break.
The match was soon turned completely on its head, though, as Logvinenko picked up the first of his two yellows for handball in the box and Fernandes tucked away from the spot.
Martins was then in the right place to steer the second home from inside the box and Doumbia finished from close range to round off the scoring, profiting from some good work from Fernandes.
The evening went from bad to worse for Astana when Logvinenko was sent off for his second bookable offence, though Ruiz sent a header wide from the best of Sporting's remaining chances.
Astana may have snatched a second late on, only for Marin Anicic's shot from range to hit the upright in a tie that is not quite over yet ahead of next week's return fixture in Portugal.
ASTANA: (4-1-4-1) Eric; Shitov, Logvinenko, Anicic, Shomko; Beysebekov; Twumasi, Maevski, Kleinheisler, Tomasov; Despotovic
SPORTING LISBON (4-2-3-1): Rui Patricio; Piccini, Coates, Andre Pinto, Coentrao; William Carvalho, Acuna; Bryan Ruiz, Martins, Bruno Fernandes; Doumbia