The former Everton and Manchester United manager had urged his players to become stronger mentally if they wanted to pick up a first away win since April and start moving up the table towards the European places.
The match was an uninspiring affair but Sociedad were on course for that victory through Sergio Canales' strike at the start of the second half before fullback Carlos Martinez handled the ball inside the area in stoppage time.
When television cameras turned to Moyes on the touchline his face was a picture of controlled anger and after Andreas Ivanschitz converted the spot-kick, he became even redder.
The decision appeared slightly harsh as the handball appeared unintentional and Moyes complained to the referee following the final whistle, but Martinez was running a risk by having his hands raised when the ball came into the box.
"We deserved the victory and it is disappointing," Moyes told reporters. "Perhaps our only error was not to have scored more goals having gone ahead."
Sociedad were missing Carlos Vela, who is their only forward of real quality since the departure of Antoine Griezmann to Atletico Madrid ahead of this season.
Alfred Finnbogason has arrived and did get a double as Real knocked lower league Real Oviedo out of the King's Cup last Wednesday but his movement has generally been too laboured in attack.
Since Moyes arrived last month, Vela and Finnbogason had been the only goal scorers and it will be an area that the Scot will target in the January transfer window.
"We need to score more. We were without our top scorer (against Levante) and this obviously made it more difficult for us," Moyes said.
"We have done well in terms of build-up play but been found lacking in the final third."
Sociedad traditionally rely on homegrown talent and it will be difficult for them to attract proven quality with limited financial resources that would help them move up the table from 14th and towards Moyes' goal of a place in Europe.
(Editing by John O'Brien)
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