Vader wanted them to get away, the Imperials planned on tracking the Millenium Falcon back to the rebel base.
Just after our heroes escape the Death Star and finish off the last TIE fighter chasing them, we get this dialogue between Vader and Tarkin:
TARKIN: Are they away?
VADER: They have just made the jump into hyperspace.
TARKIN: You’re sure the homing beacon is secure aboard their ship? I’m taking an awful risk, Vader. This had better work.
We also get this dialogue on the Millenium Falcon:
LEIA: ...they let us go. It’s the only explanation for the ease of our escape.
HAN: Easy… you call that easy?
LEIA: They're tracking us!
HAN: Not this ship, sister.
Frustrated, Leia shakes her head.
Remember while Leia was still prisoner, the Empire tortured her into giving away the base on Dantooine. The Empire checked this out, found the base deserted, and realized she'd never just give them the real location of the rebel base. Which is where the tracking scheme comes in.
Source: The ANH transcript