Martinez played a decisive role in the breakthrough early in the second half. Just seconds after making a crucial save, he launched a long pass that released Jadon Sancho, who calmly squared for McGinn to score.
McGinn told reporters after the match: "He's a mad man. I know he's a family man, but he's a big kid inside, a lively character and a top goalkeeper. Even at 2-0 he made a great save."
He added: "It was brilliant from Emi, but it highlights Jadon's performance. It was his best in a Villa shirt. He was man of the match by a mile. The manager told me to get in the box and thankfully it came to me."
Late on, Leon Bailey finished from close range after a pass from Ollie Watkins to put the result beyond doubt.
Villa dominated large spells, although clear chances were limited early on. Amadou Onana forced a strong save before the break, while Lille thought they had a way back when Olivier Giroud scored, only for the goal to be ruled out for offside.
Villa boss Unai Emery was pleased with his side's discipline across both legs. He said: "I am happy with how we performed over two matches. We were organised, we respected the opponent and defended well."
He added: "It was not brilliant, but we were consistent. When we had our moments, we were ruthless. This competition is very difficult to win, but we want to enjoy the process and keep improving."
The result extended Villa's winning run in Europe to seven matches and set up a quarter-final clash with Bologna FC 1909, as Emery targets a fifth Europa League title.