Durrant won the opening set at Alexandra Palace before the number six seed from Londonderry levelled.
Gurney trailed again as Durrant moved 2-1 in front but he fought back to take the fourth set 3-1.
Durrant edged the next two sets to progress to the last 16 on a 3-1, 1-3, 3-2, 1-3, 3-2, 3-2 scoreline.
Gurney was favourite to make the fourth round with his English opponent 21 places lower in the seedings.
It was a close contest which ebbed and flowed but Durrant held his nerve in the final two sets to go through.
Meanwhile, Fallon Sherrock’s fairytale run at the World Championship came to an end in the third round as she fell to a 4-2 defeat to 22nd seed Chris Dobey at a raucous Alexandra Palace.
The 25-year-old became the first woman to win a match in the tournament when beating Ted Evetts before following it up by stunning world number 11 Mensur Suljovic.
But Dobey proved too strong, fighting back from 2-1 down to silence the crowd and set up a clash with Glen Durrant in the fourth round.
Sherrock, who was cheered to the rafters throughout, twice led by a set, going 1-0 and then 2-1 up in the early stages, helped by some superb finishing.
Indeed, she hit 50 per cent of her doubles in the match overall, compared to Dobey’s 36 per cent.
Dobey made his scoring count, however, averaging 101.09 to Sherrock’s 90.45, while he also hit eight 180s compared to Dobey’s 11.