Three and easy for Everton
Swansea's hopes of avoiding a second successive defeat were in tatters when substitute Nathan Dyer, who had only been on the pitch for 12 minutes, was sent off in the second half.
Marouane Fellaini put the result beyond doubt with a late goal and Everton's third win of the league campaign moved the Toffees level on points with leaders Chelsea, although they are likely to surrender that lofty position by Sunday evening.
Swansea had handed a full debut to Spanish midfielder Pablo Hernandez, the club's record signing from Valencia, while Anichebe replaced injured Croatia striker Nikica Jelavic for Everton.
Anichebe puts Blues ahead
After the disappointment of squandering the lead in Monday's draw against Newcastle United, Everton looked determined to make amends in south Wales and the visitors opened the scoring in the 21st minute.
Fellaini beat Alan Tate to a high ball and the Belgian midfielder's flick, which appeared to come off his elbow, reached Anichebe, who fired home to claim his second goal of the week.
Steven Pienaar unhinged the Swansea defence with a burst down the right flank and his low cross found Mirallas, whose initial shot was touched onto the bar by Michel Vorm before the Belgian striker reacted quickest to nod the rebound over the line.
Fellaini hits third goal
Michael Laudrup's side were being carved open time and again and Mirallas' second-half shot was pushed onto the bar by Vorm, who then scrambed to block Anichebe's effort from the rebound.
There was no way back for Swansea after Dyer's foolish 57th-minute dismissal. The winger, who replaced Hernandez at half-time, had already been booked for dissent when he launched into a rash challenge on Leighton Baines, giving referee Anthony Taylor no choice but to show a second yellow card.
Everton kept pressing for more goals and Fellaini's effort was cleared off the line by Leon Britton. But Fellaini sealed the points in the 82nd minute as his header from Baines' free-kick took a deflection off Ashley Williams and flashed past Vorm.