Shaqiri’s magnificently executed chip from a sublime pass from Bojan Krkic would have been a fitting winner as he put Stoke ahead for the second time just before the break.
But Lukaku is something of an unstoppable force on his home turf and two more goals from the striker — taking his tally for the season to 18 and eight in his last seven matches at Goodison Park — cancelled out Shaqiri’s double.
That would normally have been sufficient drama for any afternoon but Gerard Deulofeu and Joselu then traded goals to set up a dramatic finish in which Marko Arnautovic earned and converted the spot-kick, slipping over as he took it, which completed a tremendous Christmas programme for Mark Hughes’ side.
“The danger today was to produce a negative performance on the back of a great performance against Manchester United,” said Hughes.
Having avoided that potential pitfall and added to their victory over United on St Stephen’s Day, Stoke must be targeting at least the ninth-placed finish they have enjoyed in the past two seasons — their best in the Premier League.
Since then Hughes has invested in further quality, which was gloriously evident as Shaqiri scored his first goals for the club.
The former Bayern Munich and Inter Milan forward opened the scoring after 16 minutes, tucking the ball away following a wonderful flowing move involving Bojan and Arnautovic.
Lukaku levelled when Erik Pieters played him onside and allowed the Belgium international to slot home but Shaqiri then produced his master-piece, clipping Bojan’s angled pass first-time over Howard as he raced clear up the inside right channel.
“It was a quality goal and he is a quality player. His CV and career bears that out. He has the potential to deliver those kinds of moments,” added Hughes.
“In recent weeks his influence on games has been really high for us. We have been delighted with what he has produced. The only thing missing, up to today, was goals.
“It’s not easy coming to the Premier League, no matter what level you’ve played at.
“But at some point Shaq was always going to make a positive impact for us and he’s starting to do that to great effect. Goals will obviously help that process even more.”
While Stoke have now lost just one of their last seven away games in the league, Everton have been guilty of letting chances slip through their grasp and did so again here.
Lukaku made it 2-2 after 64 minutes with another power-packed finish as he chested down a long ball from Tom Cleverley and fired low past Jack Butland.
Deulofeu then put the home side in front as he tapped in a Ross Barkley cross but Everton’s defencesive vulnerability was exposed again when Tim Howard flapped at a cross for Joselu to volley in his first Stoke goal with 10 minutes to go.
And then, right at the death, John Stones brought down Arnautovic as he stretched to make a tackle, referee Mark Clattenburg taking an age to make a decision that left Everton boss Roberto Martinez furious.
“We have got to be a bit wiser and we have got to learn quickly. We did not defend well enough,” said Martinez, who insisted Stones had played the ball.
“But the feeling is extra frustrating today because you can’t understand how such an experienced referee gives that decision right at the end. It’s such a wrong call.”
Howard 5; Coleman 6, Stones 5, Funes Mori 6, Galloway 5; Barry 7, McCarthy 6 (Cleverley 26, 7 (replaced by Naismith 65, 7); Deulofeu 6, Barkley 7, Kone 5; Lukaku 8.
Butland 7; Johnson 7, Shawcross 7, Wollscheid 7, Pieters 6; Whelan 7; Shaqiri 8 (Van Ginkel 87), Cameron 6 (Walters 78), Afellay 6, Arnautovic 7; Bojan 8 (Joselu 63, 7).
Mark Clattenburg 7.