The Paraguayan defender was forced to leave the field in the 32nd minute on Saturday after coming off worse in a collision with visiting striker Wilfried Bony.
And manager Roberto Martinez confirmed after the final whistle that Alcaraz, whose start to life at Goodison Park was hampered by a groin injury last season, was set for a lengthy layoff.
"He's damaged the joint in his shoulder," the Spaniard said. "We're going to assess him tonight and tomorrow morning, but it seems he's going to be out for a long spell.
"It's a real shame because he had to work really hard to get back into full fitness and being able to perform.
"He fought hard for his place and now he was in a very good moment, so it's a shame that we're going to lose him for a long spell.
"It was a bit of an accident really. You don't get those sort of injuries in football, so to get a shoulder injury at this stage of the season is very disappointing.
"We'll help him through and we hope he can come back before the second half of the campaign."
Alcaraz was lucky not to concede a penalty shortly before leaving the field when he appeared to handle from a Jonjo Shelvey strike, but Martinez insisted: "I haven't seen it."
It proved to be a frustrating 90 minutes for Everton, who failed to make their first-half dominance pay and did not capitalise on a late red card for Shelvey - booked in each half.
"It became a little bit of a frustrating game for us," added Martinez. "If you look at the stats before the sending off, Swansea had no corners or shots on target.
"It is a game that probably was less open than I expected, but we're happy with the defensive side of our performance.
"I don't think the way we were moving the ball was quick enough. I don't think we had our normal tempo in possession, and that allowed us to get frustrated."
