Gareth Bale got things rolling with the opening goal inside three minutes and Isco added a second before Cristiano Ronaldo continued his stunning form in the competition with Madrid's third just before the hour.
That saw Ronaldo equal the Champions League record of 14 goals in a season and made Madrid firm favourites to progress through and avenge last season's semi-final exit at the hands of Dortmund.
Carlo Ancelotti's men could have added even more goals but found Dortmund goalkeeper Roman Weidenfeller in fine form, while the under-strength visitors failed to find the away goal which could have put a very different complexion on things for the return leg at the Westfalenstadion.
Madrid suffered a blow just hours before kick-off when winger Angel Di Maria was ruled out through illness, but the Spanish giants did not seem to miss the Argentinian with his replacement, Isco, impressing.
Bale's early goal would also have helped.
Karim Benzema fed the ball into Daniel Carvajal inside the area and the full-back immediately played it into Bale's path, with the Wales international taking one touch before poking it past Weidenfeller.
That was just the start Dortmund, without six players including suspended star striker Robert Lewandowski, would have been hoping to avoid - and it got worse 23 minutes later when Isco made it 2-0.
Weidenfeller had already done superbly to tip over Ronaldo's free-kick but he was powerless to keep out Isco's side-footed effort after Dortmund had been robbed of possession near their own box.
With their tails up, Madrid pushed for more goals in a bid to kill the tie off even before they travel to Germany, and Weidenfeller produced another superb save to push over Bale's dipping free-kick before the Dortmund keeper denied the world's most expensive player with an outstretched foot at the start of the second half.
Dortmund briefly threatened immediately after that but Pierre-Emerick Aubameyang dragged his shot wide and Madrid then made it 3-0 just before the hour.
Luka Modric picked out the run of Ronaldo into the area and the FIFA Ballon d'Or winner showed great feet to take the ball in his stride, slip it past Weidenfeller and side-foot it into an empty net - matching the tournament record of goals in the same season.
Knowing their chances of going through were in danger of being sunk before the second leg, Dortmund pushed for an away goal but a superb block from Pepe denied Henrikh Mkhitaryan and Iker Casillas produced a brave point-blank save from Marco Reus.
But as they pushed forward, Dortmund could also have conceded more and would have been grateful to see Weidenfeller tip over Benzema's shot and see Bale blaze over after a good run.
Dortmund can take some solace from the fact they beat Madrid 4-1 at home in last year's semi-finals, and the four-goal hero of that game Lewandowski will be available again, but even a repeat of that scoreline would not be enough to send them through following this result.