Opening the Portsmouth scoring for the season, Harry Redknapp's strikeforce of Jermaine Defoe and Peter Crouch as well as defender Glen Johnson all put in a loud shout for an England place after netting against the Toffees.
Forced into fielding another young and inexperienced side, Moyes' men left the field to a chorus of boos from a frustrated Everton crowd after they were easily out-matched by a well-organised Pompey team.
Despite having the majority of the possession in the first half, Everton could not break through the towering Portsmouth side who were happy to sit deep and catch the Toffees on the break.
Everton started brightly, testing David James in the Pompey goal after four minutes with a clever free-kick form Mikel Arteta that caught the back line napping and found Yakubu in the box but the Nigerian's shot was easily smothered by a well-placed James at his near post.
The visitors seemed intent on sitting deep and getting the ball to the wing-backs and catching their opponents on the break.
The game-plan worked with the opening goal coming on 12 minutes through a predictable Crouch flick and a Defoe finish.
A long throw by Sylvain Distin found 6ft 7in striker Crouch who flicked the ball unchallenged to his partner Defoe who looked to have missed his shooting opportunity, but a clever drag-back and quick feet fooled the Everton defence and his low shot crept inside Tim Howard's goal.
The lead could have been doubled just a few minutes later after another flick-on by Crouch found Defoe in an almost identical position but his right-footed effort fell the wrong side of the same post.
Everton's best chance came midway through the first half when Yakubu's cross found Arteta six yards out, but the Spaniard's six-yard volley brought the best out of the Pompey shot-stopper who brilliantly pushed the ball onto the post before gathering the rebound.
Everton tried to get back on level terms and pressed the visitors' goal with efforts from James Vaughan, Yakubu and young Jack Rodwell but none could force James into a save.
The missed chances weighted on the minds of the Everton players and Portsmouth added to their troubles by doubling their lead on 40 minutes after a neat one-two between Johnson and Defoe easily opened the home side's back line.
A surging run by the left-back created space to lay a pass to Defoe's feet inside the box and, with no pressure, the forward had time to turn and give the ball back to Johnson who had continued his run into the six-yard box and he easily slid the ball past a stranded Howard.
Everton responded to the frustrated shouts from the crowd and almost pulled a goal back at the start of the second period when Vaughan was adjudged to have been bundled over in the box.
Yakubu hoped to take his top-flight tally to 101 goals but, after stumbling in his run-up to try and fool James, the striker hit a weak shot which was easily saved by the England number one.
Desperate for a foothold in the game, Everton threw everything they had at the Portsmouth side but long-range efforts from Phil Neville on 55 minutes and Yakubu on the hour mark never troubled the veteran shot-stopper.
The game was put out of Everton's reach on 69 minutes after a clever back-heel from Sean Davis rolled neatly into the path of man-of-the-match Defoe, who chipped the ball from outside the box onto the crossbar. It bounced down and over the line before Crouch headed home to make sure.