The Toffees suffered an early shock when Tony Hibbert was sent off for a professional foul with only seven minutes on the clock, and Sigma pushed forward immediately to try and take advantage.
But having weathered the storm, Everton hit back shortly before half-time through Steven Pienaar, whose emphatic strike effectively ended the tie and rendered Pavel Sultes' late equaliser a mere consolation.
Taking a comprehensive 4-0 advantage to the Czech Republic from the first leg, it seemed all that David Moyes' side had to do to ensure victory was guard against complacency.
But within minutes of the kick-off the Merseysiders had made life unnecessarily difficult for themselves, as Leighton Baines lost possession just metres from his own area.
Jakub Petr, who had already had one shot saved by Tim Howard, latched onto the ball and was brought down heavily from behind by Hibbert on the edge of the box.
Despite protests from the Everton players, the referee was in no doubt and immediately showed Hibbert the red card to reduce the visitors to 10 men.
Sigma looked to quickly seize the initiative, and after Lukas Bajer had rattled the crossbar direct from the free-kick, Tomas Horava and Tomas Janotka fired in an effort each within the next two minutes.
The hosts continued to dominate for the rest of the half, with Janotka in particular looking lively, but they could not convert their possession into goals and were made to pay for it three minutes before the break.
In Everton's first meaningful attack, Leon Osman broke clear on the right and pulled the ball back to find Pienaar arriving in the box. He took one touch with his left foot to control the ball before smashing it past Tomas Losavik with his right.
The goal appeared to deflate Sigma, and Everton looked comfortable after the break despite being a man down.
Osman and Pienaar were replaced by 17-year-olds James Wallace and Jose Baxter midway through the second half, while Yakubu came off the bench to make his long-waited return from injury with 15 minutes to go.
It was Sigma's substitute Sultes who managed to drill home for the hosts with 10 minutes to go, but it was too little too late for the Czechs as Everton advanced to the group stages.