Patrick Van Aanholt's free-kick opener and a brace from Lamine Kone gave the home side the win they needed to consign Norwich and north-east rivals Newcastle to the drop instead with one game of the season still to go.
It is the latest in a growing list of last-ditch escapes for his side and Allardyce is convinced they can now learn the lessons from their experience and begin to look upwards in the seasons ahead
Allardyce said :"Fire-fighting is something only this year's recruitment will put to one side
"We have been in this position three times before and said never again, and that's exactly what needs to happen this time around
"We will all sit round the table and begin the discussions with hard facts about how we're going to get where we're going to go."
Van Aanholt's 38th-minute free-kick - which left Everton keeper Joel Robles strangely rooted to his spot - shattered the mounting tension at the Stadium of Light as confidence clearly flooded back into the home side
Just four minutes later Kone volleyed home the second after Leighton Baines could only half-clear a Wahbi Khazri corner, and the defender's second early in the second half effectively sealed the Black Cats' survival
Allardyce admitted he had suffered doubts over his club's survival prospects, noting his side's 6-2 loss at Goodison Park in November, and closing 2015 with five defeats on the bounce
He said: "I remember losing 6-2 at Everton and tonight it's 3-0, and that shows how far we've come
"It is outstanding from the grim dark days of December when we lost five games on the trot and it rang home that the only chance we had to survive was to get new players in
"That's proven to be the case - since the turn of the year our performances and ultimately our results have got us clear of relegation with a game to spare and the last two games in particular have been a great joy and satisfaction."
If Allardyce's mid-season signs of the likes of Jan Kirchhoff and Yann M'Vila made all the difference, his mood was in stark contrast to his opposite number Roberto Martinez, who must be fearing for his job after this latest setback.
Despite chances for the likes of Matthew Pennington and Romelu Lukaku - who hit the bar late on - Everton were dire and undermined Martinez's post-match insistence that he had "100 per cent" belief that his players are still fighting to succeed.
Instead, Martinez bemoaned his side's misfortunes in front of goal and admitted only that the home side's extra conviction and purpose had turned the match in their favour.
Martinez said: "At the moment since the FA Cup semi-final it has been difficult for us to get that belief and intensity in the league and Sunderland were the opposite - a fully-focused team who created a good momentum because they were close to finishing the job
"We were a little bit pedestrian but I thought the effort and intent was there in a very competitive game."
Source : PA
Source: PA