Portsmouth 0 Everton 1
Leon Osman helped Everton open up a nine-point gap between them and the relegation zone with his first Premiership goal f the season.
Moyes now believes his side can start looking up and not down the table. The upbeat Scot said: "To win three games in a row at this stage of the season is no mean achievement and we have got to try to finish in the top half now.
"It was a well deserved victory and we have got ourselves on a bit of a roll.
"Some results were good for us today and some were not, but you have got to be positive.
"We will try to get as many points between now and the end of the season as possible.
"Once we went ahead, I did not see any way Portsmouth were going to get back in the game and I don't think Nigel Martyn had too many saves to make."
Pompey boss Harry Redknapp has shelled out £11million of new co-owner Alexandre Gaydamak's personal fortune since the transfer window opened a fortnight ago.
Redknapp handed immediate debuts to newly acquired Spurs trio Noe Pamarot, Sean Davis and Pedro Mendes as well as £ 4.1million record signing Benjani Mwaruwari. But none of the newcomers managed to impress in a Pompey performance that summed up why they are staring relegation in the face.
In fact Everton keeper Martyn was called into meaningful action only twice in the match.
Matchwinner Osman should have scored inside 60 seconds. Tim Cahill broke free on the right and put in a teasing cross only for Osman to head over from close range.
Mendes attempted to mark his debut in spectacular fashion with a 40-yard lob that had Martyn struggling after full-back Matteo Ferrari's poor clearance.
Martyn was called into action again a minute later when he superbly beat away Lomana LuaLua's strike at the near post after LuaLua and new strike partner Mwaruwari had linked up on the right.
Mendes picked up the first of four yellow cards of the game in the 26th minute for a foul on Philip Neville as a first game memento.
The Portuguese international and his fellow new boys' afternoon got worse in the 31st minute when Osman popped up to scoop home the winner.
Joseph Yobo's long-range free-kick was flicked on by Republic of Ireland winger Kevin Kilbane and striker James Beattie bulldozed his way past two defenders.
The ball fell kindly for Osman, whose scuffed shot was helped into his own net by home defender Andy O'Brien.
Cahill could have doubled the lead nine minutes later when he headed over from an Osman centre.
The Australian international had another opportunity to put the game beyond doubt on the stroke of half-time when his downward header was acrobatically tipped over the crossbar by Pompey goalkeeper Jamie Ashdown.
Davis was given a debut yellow card early in the second half when he booted the ball away after a free-kick was awarded against him, as the home side's frustrations began to show.
The busy Osman came within inches of doubling his and Everton's tally on the hour mark when he shot just wide from 25 yards.
Redknapp threw on Poland striker Emmanuel Olisadebe, another new signing backed by Gaydamak's millions, in place of the ineffective Richard Hughes, soon after.
But the on-loan frontman made little impact, in keeping with the other new boys on show.
Ferrari and Kilbane both added their names to the yellow card count before Davis wasted Pompey's best opening of a dire second half.
Andy Griffin's curling free-kick was only half cleared, but Davis let Everton off the hook when he blazed over the crossbar in a moment that summed up his afternoon.
Davis, Pamarot and Mendes were all deemed not good enough by Tottenham boss Martin Joel and will have to play much better than this to prove the Spurs boss wrong.
Gaydamak may have to delve into his pocket again with LuaLua and Benjani both heading off to play in the African Nations Cup.
Redknapp said: "It will be a blow to lose them but what can you do about it? A lot of clubs are in the same position and we just have to get on with things."