The Germany starlet admits he should have done better with his one clear-cut chance earlier in the clash but feels the Spanish giants were too strong on the night
Two goals each for Cristiano Ronaldo, Karim Benzema and Gareth Bale saw Carlo Ancelotti's side secure an all-but-unassailable advantage in Gelsenkirchen, despite Klaas-Jan Huntelaar's emphatic late consolation.
Draxler was thwarted by Iker Casillas from point-blank range in the first half with the scores at just 1-0, but the young Germany international believes his side was punished for a number of mistakes on the night.
"We had high hopes but in the end we committed too many mistakes and Real took advantage of it," he told Sky after the match at the Veltins Arena.
"Yes, of course I had to convert that [first-half chance]. It got a little deflection but that one has to be considered a sitter. But when you lose 6-1 it is difficult to talk about what could have been.
"Keller told us at halftime we were unlucky at the goals and how we should try and score one or two to have a little chance in the second leg. But Real got more space and they totally took advantage of that.
"You can't blame us for not fighting. I think we gave everything. But against Real it is very difficult to get into the duels, because they move the ball so fluently. We were always one step too late."
The second leg takes place at Santiago Bernabeu next month.