There were three red cards in the contest, with Elche captain Gonzalo Verdu dismissed late in the first half before Mario Hermoso was sent off for the hosts early in the second.
Joao Felix and Morata struck after that to seal the win, with Elche substitute Domingos Quina also seeing red in stoppage-time as the Rojiblancos eased to three points at Metropolitano.
After a three-game winless streak ahead of the mid-season FIFA World Cup break checked its momentum, Diego Simeone's side will be relieved to get back to winning ways.
Chances were few and far between for Atletico in the first half, and it was telling that even Verdu's video assistant referee-assisted dismissal for an off-the-ball foul on Morata just before the break did little to lift the hosts.
Aleti's man advantage was temporary too, with Hermoso handed his marching orders after back-to-back bookings, the second for a reckless foul on Helibelton Palacios after the interval.
But Joao Felix's close-range header, turning in Antoine Griezmann's short rebound cross, gave a fresh lease of life to the hosts and allowed them to play with greater freedom.
That translated into a second goal, this time after Morata exquisitely dribbled in off the right wing to turn the defence inside out in the 74th minute, and effectively put the game to bed with a deflected effort that lopped over Edgar Badia and in.
Elche's efforts to respond saw their discipline slip once again, with late substitute Quina also dismissed for a quick double booking, the second after a clumsy foul on Morata, as Atletico cruised to victory.