The former Chelsea striker scored in the 70th minute for the team from the Spanish capital, but a late thunderbolt strike from Lucas Perez handed Atletico Madrid their fourth draw in five La Liga matches.
The 27-year-old has scored three goals in his last three matches after previously scoring one in seven games.
The former Real Madrid and Juventus forward admitted that he is gutted about the poor form of Atletico Madrid, playing down his own return to form.
"If I worried about myself I would play tennis. I'm as disappointed as everyone else is in the locker room," Morata said after Atletico Madrid’s stalemate against Alaves.
"We had chances and we did not take advantage of them. For us, two points escaped us, but we also have to see the rest of the teams and what they will do on a trip as difficult as this one."
The Spain international had an opportunity to increase the visitors’ advantage shortly before Alaves’ equalizer and head coach Diego Simeone refused to blame Morata but rather praised Fernando Pacheco for a great save.
Simeone said: "The goalkeeper made a great stop, the ball was not going wide of the goal,"
"Alvaro was fine and the goalkeeper kept Alaves in the game because the match was over with the second goal."
