A late Mats Hummels strike in the reverse fixture a fortnight ago gave Dortmund two away goals and a slender advantage heading into their last-16 clash at Westfalenstadion.
Following an incident-free opening half hour, Santana headed Dortmund clear in the tie with a powerful header from Gotze's corner.
The Germany international soon added a goal to his assist when he found space in the box to calmly convert from Robert Lewandowski's cross.
Lewandowski's Poland teammate made the result emphatic just prior to the hour mark, when Andriy Pyatov spilled a long-range shot and allowed Blaszczykowski to tap in the rebound.
Despite being all but out of the contest, Shakhtar enjoyed their best spell of the match in the final 20 minutes.
Luis Adriano, Fernandindo and Yaroslav Rakitskiy were all denied by home keeper Roman Weidenfeller as the visitors failed to find a consolation goal.