Atleti were winless in three in all competitions and four points off the pace in LaLiga coming into Sunday's fixture, with leaders Barcelona and second-placed Real Madrid each winning a day earlier.
Diego Simeone's men fell a goal behind to Sergi Darder's clinical strike, before Morata came to the fore, creating Angel Correa's equaliser and then scoring himself.
Koke made sure of the points in stoppage time to send Atleti into the international break on a high after a mixed start to the season.
⚽️ vs. Bayer Leverkusen— Squawka Football (@Squawka) November 6, 2019
⚽️ vs. Athletic Club
⚽️ vs. Alaves
⚽️ vs. Sevilla
⚽️ vs. Bayer Leverkusen
Alvaro Morata has now scored in five straight games for Atletico Madrid. pic.twitter.com/02SLoLDufc
Even as the team have toiled, though, losing to Bayer Leverkusen in the Champions League in midweek, Morata insists their attitude cannot be questioned and the squad are united in ignoring external analysis.
"We are always motivated," he said, having scored in a sixth straight game in all competitions. "It may be that sometimes we do well and sometimes we do not. We always go out to win - and today, we have done it.
"Inside the team, we are a compact group and we are not affected by what is said about us. If we do what we have to do, we will go far this season."
Morata added he felt Atleti had been unfortunate in previous matches not to win and so had no qualms about collecting three points from an unconvincing display against struggling Espanyol.
"The important thing is that we win," he said. "I always say it.
"In the end, it is about being all together, in moments like the other day [in Leverkusen] or in Seville [last week's 1-1 draw with Sevilla], matches in which I think we deserved more.
"But today we have won and added three important points to move up the table.
Morata's contributions turned the match on its head after a tough start in which Atleti looked uncharacteristically shaky at the back, not aided by the unorthodox presence of Saul Niguez at left-back.
Renan Lodi, talked up by his coach on Saturday, was left on the bench, and Simeone explained: "I wanted to have more strength and to have a replacement for Lodi, who had been playing many games and needed a break."