The Pitmen ran away with a 4-2 victory in the Turkish capital, putting themselves in a good spot for the return leg next Thursday in Krakow. It took them just seven minutes to go in front through Alisson, as the Brazilian winger beat several opponents before sending Mert Gunok the wrong way.
Eguinaldo added a second just before the half hour mark, putting the Ukrainian giants two to the good. A few minutes before the break, Tammy Abraham got one back for the home side from the spot.
Arda Turan's men continued their solid display with a 51-pass series that led to Kevin netting their third. The former Palmeiras man added a fourth in stoppage time after Joao Mario scored a second 10 minutes prior.
Shakhtar, who are now 9-1-0 versus Turkish sides, have scored 10 times in Europa League qualifying this summer, more than any other team.
Elsewhere, Eredivisie side Utrecht overcame an early deficit to prevail 3-1 versus Sheriff Tiraspol. Souffian El Karouani assisted Victor Jensen and Nick Viergever to give his side the lead, before Adrian Blake added a third 20 minutes from time.
There were two matches that ended scoreless. The UEL's ever-present side, Sporting Braga, were held in Bulgaria against Levski Sofia, For the first time since the competition's inaugural season under its current guise, the Portuguese outfit were held off the board in a qualifier on the road.
The other goalless encounter occurred in Switzerland, as CFR Cluj's Karlo Muhar failed to convert a penalty 11 minutes in against Lugano. While the home side had most of the ball, they could not convert any of their 11 attempts into a goal.
Anderlecht were the only other side to win on the night, thanks to a Kasper Dolberg strike 10 minutes before the interval. That was enough to give them a narrow 1-0 success over Hacken.
Doguhan Simsir's 72nd minute goal cancelled Jamie McGrath's opener for Hibernian as they were held 1-1 in Denmark versus FC Midtjylland. As for Banik Ostrava, they twice held the lead against Legia Warsaw, but were pegged back twice in their 2-2 draw. Michal Frydrych thought he scored his second of the match at the death, but it was ruled out for offside.
In Cyprus, Celje and AEK Larnaca could not be separated, as the match ended 1-1. Both teams got their goals via a penalty, with the Slovenians finding their equalizer eight minutes after the restart.
All eight return legs are slated for next Thursday, with the winners making it to August's third qualifying round. Joining them there will be Wolfsberger, Fredrikstad, and PAOK, along with the 15 losing teams from the Champions Leaue second qualifying round.
Follow all of the UEFA Europa League qualifying matches on Flashscore.