The Whites had trailed to Alan Browne’s 22nd minute opener and looked like they would be left with nothing to show for their dominance.
Preston had been forced to soak up a wave of pressure since taking the lead.
Ezgjan Alioski missed an open goal, before he hit the post and then saw a late header brilliantly kept out by Declan Rudd.
But their pressure was finally rewarded when Dallas slotted home from just outside the box with the aid of a deflection off Ryan Ledson.
It leaves Leeds eight points clear of nearest chasers Brentford.
Brad Potts scuffed an attempted long range shot in the seventh minute before Tom Barkhuizen had a shot saved from a tight angle in the 15th.
Browne shot wide minutes as later as Preston continued their confident start.
And they got their reward after 22 minutes when Browne calmly slotted home after a brilliant counter attack
But plenty of credit will also go to Sean Maguire for a fantastic tackle which robbed Mateusz Klich of the ball to set the attack going.
Alioski shot wide in the 28th minute, before he and Luke Ayling both fired over as the hosts began to show more urgency.
A stretching Ayling could not direct a header in the 44th minute as Leeds still waited to register their first shot on target.
Maguire fired over seconds after the restart following a careless pass out of defence by Liam Cooper..
Barkhuizen had a shot blocked for a corner by Ben White in the 49th minute after Preston again hit United on the break.
Ben Davies was forced to turn the ball behind for a corner in the 52nd minute as Leeds pushed North End back.
Rudd beat away a powerful Dallas shot at his near post, before the Leeds winger scuffed a follow-up effort wide.
Maguire smashed an effort well over on the hour mark after he got between two defenders and created a half-chance.
Alioski somehow fired over an open net on 65 minutes after Helder Costa’s cross from the byline caught Preston napping.
Rudd tipped away Eddie Nketiah’s glancing header a minute later as the intense pressure continued to grow.
Alioski smacked an effort against the foot of the post in the 75th minute from just outside the box.
His luck was out again when Rudd pulled off a superb reflex save to keep out his header.
But Rudd was finally beaten when Ayling kept the ball alive to set up Dallas to give Leeds a deserved leveller.