Carlos Vela opened the scoring for Sociedad from the spot but Oscar Diaz levelled for the hosts before the break.
It set up an action-packed second half, which saw Imanol Agirretxe restore the visitors' lead, only for Verza to score two penalties in quick succession to put Almeria ahead for the first time.
Markel Bergara looked to have grabbed a point for La Real when he headed home with five minutes remaining, but a fortuitous goal by substitute Hicham Khaloua in injury time gave Almeria all the points.
The first controversial moment arrived in the 20th minute, with Angel Trujillo penalised for blocking a through ball in the area with his hand, although the defender argued the movement was not intentional.
Nonetheless, Vela kept his cool and wrong-footed Almeria goalkeeper Esteban to open the scoring.
Perhaps spurred on by their sense of injustice at going behind in the manner they did, Almeria responded with a superb goal eight minutes later.
Diaz played a short pass to Fernando Soriano, who delivered a penetrative through ball over the top for his team-mate to run onto and smash past Claudio Bravo, to the delight of the home fans.
Antoine Griezmann should have restored La Real's lead seven minutes into the second half, firing just wide of the far post with just Esteban to beat, after Agirretxe had released him with a poked pass.
Almeria then scored what looked to be a perfectly legal goal but it was ruled out, Verza harshly judged to have been offside when he knocked Soriano's downward header into the net.
Their sense of injustice was magnified when La Real went up the other end and scored in the 58th minute, Agirretxe executing a delicious chip over Esteban just as he trod into the area.
But Almeria were level soon enough, Verza slamming home a harshly awarded penalty after Carlos Martinez fouled Aleix Vidal a couple of feet outside the area in the 66th minute.
The action was far from over. La Real came centimetres from taking the lead for the third time, Almeria defender Mane sliding in to scrape Agirretxe's shot off the line just as the ball was about to roll into the net.
The hosts then attacked, Vidal racing through the middle and into the box, falling to the floor once his shirt was tugged by Mikel Gonzalez.
Verza stepped up again to drill the ball into the top corner in the 71st minute and give Almeria the lead for the first time in the game.
But the hosts' joy did not last long, Bergara appearing alone at the far post to head a free-kick into the far corner of the goal.
However, Almeria would have the last laugh. An attack from the hosts looked to be about to come to an end when Alberto de la Bella stepped up to clear the danger, but booted the ball off the thigh of Hicham.
Almeria coach Francisco Javier Rodriguez's frustration with referee Miguel Angel Ayza Gamez boiled over in injury time and he was sent off for disputing another decision.