The same pattern of relentless home pressure continued after the interval, with Saha the biggest threat.
Saha dispatched the penalty with ease to set up a jittery finale for Hull.
Everton went close through a curling effort from Steven Pienaar, while Rui Patricio saved from Louis Saha.
Wriddhiman Saha followed shortly afterwards, also edging behind but this time off the bowling of Peter Siddle.
Louis Saha gave the Toffees the lead when he headed past Petr Cech from Leighton Baines' right-wing corner.
Mr Saha is the fourth journalist to have been killed in the region over the last three years.
Zayatte conceded a penalty which Louis Saha converted for the visitors but the Tigers clung on for victory.
Louis Saha broke the deadlock when he steered the ball past David James.
Louis Saha, Victor Anichebe and James Vaughan were all struck down by injury.
Saha was the player who took the Dutchman's place in the starting XI for the Carling Cup final against Wigan.
Louis Saha put Everton ahead from 20 yards early on but Chris Dagnall headed wide when well placed to equalise.
But Saha's fifth league of the goal of the season after 42 minutes soon changed the mood at Fratton Park.
The Toffees, without their top scorer Tim Cahill, relied on Saha, Rodwell, Marouane Fellaini and Diniyar Bilyaletdinov to provide the bullets.
But he had left the ground by the time the team ran out and Giuseppe Rossi was selected to play alongside Saha.
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And Moyes reserved special praise for match-winner Saha, who he tipped to have a big impact on the run-in of Everton's season.
Trabelsi carelessly played Dunne into trouble, and when he was robbed by Rooney, Gabriel Heinze crossed for Saha to bundle past Weaver.
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It was a poor spot kick from Saha, whose effort lacked power and was struck at a good height for the keeper.
Everton could have extended their lead before the break, when Saha met a Baines free-kick but planted his header well over the bar.
Louis Saha missed a gilt-edged chance when Wigan keeper Ali Al Habsi blocked his strike before Tim Cahill's header hit the Latics post.
United do have Fraizer Campbell and Danny Wellbeck who are young and inexperienced, while Louis Saha has been bedevilled by a succession of injuries.
Saha was again guilty of taking too long as he was put through on 66 minutes, allowing Andrew Corbett to make a good tackle.
The former premier, back as chief of the Saha Union industrial group, envisions his country as the information technology (IT) hub for the Mekong basin.
Boro led 1-0 at half-time through David Wheater's 44th-minute goal but Marouane Fellaini and Louis Saha scored to give Everton a showdown against Manchester United.
Teenager Jack Rodwell scored two fine goals and Louis Saha also notched a brace as Everton look set for the group stages of the competition.
Osman thumped home his header to give his side a lead they might have had much earlier had Louis Saha not blazed over when well placed.
Saha missed a glorious chance to add United's third after 52 minutes, but he shot wildly over the top after Isaksson could only block Rooney's shot.
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Everton finally hit back in injury-time as striker Louis Saha, who replaced midfielder Leon Osman in the second half, netted after Steven Pienaar's shot was blocked.
Key forward Louis Saha will be out slightly longer, with Ferguson confirming it would be mid-August before he will be back on the training field at Carrington.
But it was Everton who dominated in terms of chances, with Joleon Lescott hitting a post and Louis Saha and Tim Cahill wasteful in front of goal.
Bilyaletdinov should have given Everton the lead two minutes later but he failed to find the target from close range after Saha dummied another dangerous cross from Baines.
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