Richarlison, Gylfi Sigurdsson, Lucas Digne and Theo Walcott all got on the scoresheet as Everton moved into seventh position in the standings, while United remain in sixth place after a third top-flight defeat in five matches.
Both Marco Silva and Solskjaer would have demanded a positive response after poor performances over the past eight days, but it was Everton who completely dominated the opening 45 minutes.
David de Gea made a terrific save to keep out a close-range volley from Richarlison, but the United goalkeeper was powerless to prevent the Brazilian from handing Everton the lead two minutes later.
United almost found an equaliser when Marcus Rashford fired narrowly over from 15 yards out, but it was a rare moment of joy for the visitors as they failed to get to grips with Everton in all departments.
The Toffees' second goal arrived when Sigurdsson was provided with too much space by Victor Lindelof, and the Iceland international capitalised by drilling a low 25-yard strike into De Gea's right-hand bottom corner.
Everton maintained control of the contest without building on their second goal, although Richarlison did have a speculative effort from distance saved by De Gea at his near post.
Solskjaer - almost inevitably - made changes at the break with Ashley Young and Scott McTominay being introduced into the action, but Everton were also forced into an alteration when Richarlison was unable to shake off an injury to his ribs.
United were doing their best to find a way into the contest, but Everton netted their third before the hour mark as Digne fired a low volley through a pack of players into the bottom corner from 25 yards.
Sigurdsson soon went close to adding a fourth when his corner had to be cleared off the line by De Gea, but it was not long before the home side further extended their advantage.
Walcott was sent clear of the United defence down the left flank, and the substitute was able to drift inside before slotting a low shot under De Gea from eight yards out.
Everton continued to push for goals during the closing quarter of the game, but United twice went close to consolation efforts with Anthony Martial and Nemanja Matic failing to hit the target from the centre of the penalty area.
Martial also had a long-range strike kept out by Jordan Pickford but any goal for Solskjaer's side would have been harsh on Everton, who have now registered 10 points from home fixtures with Liverpool, Chelsea, Arsenal and United.
EVERTON (4-2-3-1): Pickford; Coleman, Keane, Zouma, Digne (Jagielka 85'); Schneiderlin, Gueye (McCarthy 76'); Richarlison (Walcott 51'), Sigurdsson, Bernard; Calvert-Lewin
MANCHESTER UNITED (4-3-3): De Gea; Lindelof, Smalling, Jones (Young 46'), Dalot; Pogba, Matic, Fred (McTominay 46'); Rashford (Pereira 77'), Lukaku, Martial