It was the first time in 43 years this fixture has produced a goalless draw.
Both sides struggled to create clear-cut chances before the game suddenly sparked into life shortly before half-time when Preston keeper Freddie Woodman palmed Ryan Hedges’ shot towards Tyrhys Dolan, whose half-volley hit the post.
Greenwood was red-carded a couple of minutes later but his team nevertheless finished the half strongly.
Brad Potts’ shot brought a near-post save from Rovers keeper Aynsley Pears, who then did superbly to keep out Andy Hughes’ header from the resulting corner.
Blackburn boss John Eustace made an attacking change at the break in an attempt to capitalise on the numerical advantage, sending forward Makhtar Gueye on in place of midfielder Baker.
Gueye saw a tame effort gathered by Pears, who was largely comfortable as the hosts defended well.
And Beck’s frustration got the better of him when he was shown a red card for reacting angrily to a challenge, kicked out at Holmes, and then claimed he had been bitten by North End substitute Milutin Osmajic - who received a yellow card - in the melee which followed.
Pears was called into action twice in the final stages, saving strikes from Ben Whiteman and Potts but Blackburn held on for a point.