Gameiro picked up where he left off after grabbing a late equaliser in the first leg last week, firing a ninth-minute opener at the Ramon Sanchez Pizjuan to set his side on course for a 5-3 aggregate win.
Shakhtar briefly drew level through Brazilian Marlos, but Gameiro’s second and a third for Mariano ensured the extraordinary continental run of Unai Emery’s side would continue.
The form guide had suggested Shakhtar had a shot at victory with Sevilla limping into the game on the back of an indifferent 1-0 loss at Espanyol last weekend, while the visitors thrashed domestic rivals Dynamo Kiev 3-0.
But as two consecutive Europa League crowns prove, Sevilla are a different proposition in Europe and it did not take them long to grab the lead on the night when Gameiro seized on a mistake by Maksim Malyshev to fire home.
There was plenty of early frustration evident in the Shakhtar ranks with Marlos, Yaroslav Rakitskiy and Oleksandr Kucher all booked in the first half-hour as their side struggled to get a grip.
Only a superb interception from Taras Stepanenko prevented a golden second chance for Gameiro, but Shakhtar slowly grew in confidence and a near-miss from Ismaily served as a warning for the hosts.
Moments before half-time Shakhtar made their pressure pay when Marlos played Eduardo into the box and the Croatian fired his side level – and still had time to miss the chance to add a second before the break.
Shakhtar’s relief lasted just two minutes into the second half before the dangerous Ever Banega threaded a neat ball into the box from out on the right and Gameiro kept his cool to round goalkeeper Andriy Pyatov and make it 2-1.
Sevilla began to pour on the pressure, with Mariano testing Pyatov and Banega firing wide before the increasingly inevitable third arrived in the 59th minute.
Shortly after Gameiro had put wide with a acrobatic over-head kick Mariano sealed the game.
The full-back rifled home a super finish from a Vitolo through-ball from just inside the box.
Grzegorz Krychowiak almost made it four when he flashed a header wide from Banega’s free-kick 10 minutes later, while Eduardo also came close to what surely would have been a consolation for the visitors.
Shakhtar’s frustrations spilled over in the final minutes with Stepanenko in particular fortunate to avoid a red card after a pair of tasty challenges, as Sevilla cruised into their third straight final in style.