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 UEFA Champions League record of 14 goals in a season and made Madrid a firm favourite 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 visitor 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 Angel di Maria was ruled out through illness, but the Spanish giant did not seem to miss the Argentina winger with his replacement, Isco, impressing.
Bale's early goal would also have helped.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Karim Benzema fed the ball into Daniel Carvajal inside the area and the fullback 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 its own box.
With its tail up, Madrid pushed for more goals in a bid to kill the tie off even before it travels 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 its 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 it 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 it 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 it through after this result.
