The Czech Republic striker scored five times in a first-leg demolition of Levadia in the last round, and the captain led by example against Malmo by helping himself to a hat-trick at the Generali Arena as Sparta took the upper hand in the tie.
Malmo went ahead courtesy of a deflected Emil Forsberg strike after 17 minutes and were in front for a second time when Isaac Thelin struck following a quick equaliser from Lafata.
The Swedish champions were pegged back again when Lafata levelled six minutes into the second half, and Radoslav Kovac put the home side in front for the first time just a minute later.
And Lafata then had the final say, as he completed his hat-trick 20 minutes from time to give the Czech champions a two-goal cushion heading into the second leg at Swedbank Stadion next Wednesday.
Elsewhere, Karol Meszaros came off the bench to score the winner five minutes from time as 10-man Slovan Bratislava beat Sheriff 2-1 at Stadion Pasienky.
Igor Zofcak put Slovan in front from the spot three minutes before half-time, but the home side had Richard Lasik sent off for a foul on Radu Ginsari that resulted in a penalty that Cadu was unable to convert as Dusan Pernis saved his spot-kick.
Ricardinho looked to have spared his fellow Brazilian's blushes when he equalised 16 minutes from time, but Meszaros had the final say when he popped up with a winner with five minutes remaining.
In Tuesday's other Champions League game, Ibrahima Sidibe scored the only goal of the game from penalty spot as Debrecen claimed a 1-0 first-leg lead against BATE.
