Everton inflicted Norwich's latest defeat as goals from Andros Townsend and Abdoulaye Doucoure saw the Premier League basement boys beaten 2-0 at Goodison Park.
Townsend put the hosts in front with a 29th-minute penalty, awarded after referee David Coote watched replays back pitchside of Ozan Kabak's challenge on Allan.
Norwich midfielder Mathias Normann subsequently brought two good saves out of Jordan Pickford before Doucoure's finish doubled Everton's advantage in the 77th minute.
The result made it six defeats from six for Daniel Farke's rock-bottom Canaries so far this term, with their losing run in the Premier League, including the final games of the 2019-20 season, now standing at 16 matches.
Rafael Benitez's Everton, returning to winning ways after their 3-0 loss to Aston Villa last weekend and Carabao Cup exit at QPR, are up to fifth in the table with 13 points from their six games.
While the Toffees had the fit-again Pickford back in goal, they were still without a number of players due to injury, including forward pair Dominic Calvert-Lewin and Richarlison.
The opening stages of the contest saw the hosts have plenty of the ball but struggle to show much cutting edge.
The first real attempt on goal came in the 19th minute when Alex Iwobi delivered the ball from the left and it went via Salomon Rondon to Townsend, who fired a shot straight at Tim Krul, and Demarai Gray then sent a cross flashing across the Norwich area that no-one could apply a killer touch to.
Soon after, Allan went down in the box having been caught on the leg by former Liverpool loanee Kabak and appealed for a penalty, with Coote initially saying no.
But a VAR check followed, Coote watched the incident back on a pitchside monitor and pointed to the spot, and Townsend subsequently produced a cool finish from 12 yards to put Everton in front.
Norwich responded towards the end of the first half with a shot wide from Kenny McLean and then a Normann strike that Pickford turned behind.
Iwobi and Townsend hit efforts off target early in the second half before Pickford did well once more to deny Normann, pushing the Norwegian's drive over the bar.
Norwich threatened again as McLean sent a free-kick towards Ben Gibson, who was just unable to make the connection he wanted as he tried to head in, and Normann then fired over a free-kick of his own.
It was making for a nervy time for Everton but that feeling was then settled when Gray teed up Doucoure, who slotted past Krul.
There could have been further misery for Norwich, with substitute Tom Davies seeing a shot deflect wide and Ben Godfrey, playing against his old club, then being denied from close range by Krul.
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