The Bees snatched a point when substitute Bradley Wright-Phillips pounced in the final minute after goals from United's Barry Robson and Dave Kitson cancelled-out Marcello Trotta's opening spot-kick.
By that stage, red cards for Tony Craig and Clayton Donaldson left Brentford with nine men and United's Harry Maguire was also sent for an early bath.
Craig and Maguire's dismissals were immediately followed by missed penalties from Tony McMahon and Sam Saunders.
"I learned a lot about my players out there," Morgan said. "I learned they have heart and desire.
"I learned they have the character and fight to come back from difficult situations.
"I firmly believe that we will go into the final match of the season still with a chance of automatic promotion.
"People might think I'm wrong but I'm of that opinion without a shadow of a doubt."