But Saha's third levelled the scores and he completed the scoring with his eighth goal in his last six games after a superb Beckford volley.
Saha had opened the scoring after 20 minutes and was angry to have another disallowed before half-time after Blackpool equalised when Alex Baptiste made sure Ian Evatt's effort crossed the line.
However, as has also been their way recently, Blackpool let their opponents back in and are now slipping alarmingly towards the relegation zone.
Blackpool boss Ian Holloway rang the changes as he tried to prevent a run of six defeats in seven league outings worsening.
Goalkeeper Richard Kingson was the biggest casualty of Wednesday's loss to West Ham as Paul Rachubka came in for his first league start of the season.
New signings James Beattie and Puncheon made their full debuts, while Evatt and Elliot Grandin returned.
But it was an unchanged Everton, with Tim Cahill available again but on the bench, who made the early running as the rain set in from the start.
John Heitinga tested Rachubka when he robbed Grandin and tried his luck from distance, but the American saved comfortably.
Everton deservedly took the lead after 20 minutes as Diniyar Bilyaletdinov weaved his way into the area and picked out Saha, who confidently swept home from seven yards.
It was over half an hour before Blackpool got a serious sight at goal but Adam shot well wide from 30 yards.
Adam got another chance after Sylvain Distin was caught out by an awkward bounce and handled on the edge of the area. Adam's free-kick was blocked but the ball ran out for the corner which led to the 37th-minute equaliser.
Adam picked out Evatt at the near post and his shot may have crossed the line after striking the woodwork before Baptiste followed up to make certain.
Everton thought they had reclaimed the lead two minutes later when Saha touched the ball past Rachubka from a Seamus Coleman pass.
However, referee Kevin Friend had already blown his whistle for a foul on Coleman just outside the area.
Jack Rodwell fed Baines down the left and the full-back found Saha with a perfect ball into the area which was turned in at the near post with a first-time shot.
Baines almost added another soon after with a low drive from the edge of the area which he just dragged wide.
Rodwell looked certain to score after a brilliant touch took him past Baptiste in the area but he flicked his shot across goal.
Everton paid for those misses as Blackpool turned the game on its head with two goals in the space of three minutes.
The second equaliser came in the 61st minute after Heitinga lost the ball to Grandin 30 yards out, who quickly fed former Everton striker Beattie down the left and he cut inside to cross for the inrushing Puncheon to sidefoot home.
Before Everton had time to recover, Blackpool broke from deep to set up DJ Campbell on the edge of the area.
Campbell's shot was tipped onto the bar by Howard but Adam followed up to dive forward and head home the rebound.
A compelling game almost took another twist when Rodwell caught Rachubka off his line, but Evatt got back to make a brilliant acrobatic clearance.
Everton got themselves back level 14 minutes from time when Saha completed his hat-trick from a corner when he headed home from close range after a Distin flick-on.
The crowd called out for another and substitute Beckford obliged in the 80th minute when he raced onto another superb Baines cross and hit a fine left-foot volley past Rachubka.
Given what had gone before, Everton did not want to take any chances and surged forward to put the game beyond doubt with a fifth.
Saha showed some of the determination of old as he beat Adam on a race to goal and slid in his fourth.