Gayle slotted home Matt Ritchie's cross from close range after 24 minutes to give Rafa Benitez's side - who carved out a host of opportunities in the first period - a deserved advantage.
But when Robbie Brady went down under the challenge of Yoan Gouffran late in the first half, referee Peter Bankes pointed to the spot, and Dorrans stepped up to level the scores at 1-1, before Cameron Jerome's curling strike six minutes after the restart turned the game on its head.
Murphy bagged Norwich's third shortly after, making it 3-1 to the visitors, with the game seemingly beyond the Magpies.
But Gayle's second of the night gave United hope, before Gouffran's header in added time looked to have clawed back a point.
And Gayle had the final say, turning in the box before dispatching low past McGovern to send St. James' Park wild and snatch all three points.
Goalkeeper Karl Darlow was handed his first league start of the season by Benitez, having kept two clean sheets in his two EFL Cup outings so far this term.
And midfielder Jack Colback came into the side at the expense of Isaac Hayden in the only other change from the team that drew 1-1 at Aston Villa on Saturday.
Alex Neil's Norwich, meanwhile, included former Magpie Sebastien Bassong in their starting lineup, as they looked to regain top spot in the Championship from Huddersfield Town.
After an uneventful opening 10 minutes, it was Newcastle who created the first chance. Jonjo Shelvey, nominated for the PFA Fans' Player of the Month earlier in the week, picked out DeAndre Yedlin on the right, and his volleyed first-time cross fell kindly for Gayle. The forward's shot, though, was parried by Michael McGovern, and the visitors scrambled it clear.
A few minutes later, Gayle had another sight of goal when he controlled a looping clearance from Ciaran Clark perfectly before dragging his shot just wide of the upright.
Gayle then nodded a Ritchie corner goalwards, only to find McGovern again equal to it, as the Northern Ireland keeper clawed it away with United firmly in the ascendancy.
After 20 minutes, they came even closer. Paul Dummett's enticing delivery from the left found Ritchie at the far post, but the winger's header cannoned back off the crossbar with McGovern beaten.
Shelvey pulled one wide from a Gayle lay-off as the one-way traffic continued to flow and, 24 minutes in, the inevitable goal finally came.
Yedlin played in Ritchie down the right, and his superb low cross with the outside of his left foot found Gayle in the area, and the ex-Crystal Palace man tucked home his fifth goal of the season from close range in front of the Gallowgate End.
The dangerous Gayle went close again, firing at goal from a tight angle after Shelvey's lofted pass found him in space, but McGovern saved with his legs at his near post.
But the Canaries, who had offered little going forward up to this point, were afforded the chance to draw level just a couple of minutes before the break. Brady collected a Dorrans pass on the right and, as the Irishman burst into the box, referee Peter Bankes adjudged Gouffran to have pulled him down as he prepared to shoot.
And Dorrans stepped up to take the spot kick, and coolly converted, sending Darlow the wrong way to send his team in level.
Five minutes after the break, Dorrans fired one just wide of Darlow's post as Neil's side began in a more assured manner this time around. It told, too, barely a minute later, when Murphy's low cross found Jerome at the far post. His curling effort from an unfavourable angle beat Darlow and nestled in the corner of the net, leaving Benitez's charges trailing.
From there on, United struggled to create as many chances in front of goal as they had in the first half. Norwich, though, looked a threat on the counter, and it was from one such break forwards that the visitors extended their lead. Murphy surged through a gap in the middle and took aim, and a cruel deflection saw the ball fly past Darlow's despairing dive, making it 3-1.
But the Magpies weren't finished there, and Gayle pulled a goal back almost immediately when his excellent control allowed him to bring down Shelvey's superb long pass and slot past McGovern to cut the gap to one goal.
With Aleksandar Mitrovic now on in place of Mo Diame, Newcastle looked to pull themselves back onto level terms. Shelvey fed Gayle over the top, and his powerful left-footed shot had to be tipped round the post by McGovern before Mitrovic headed a Ritchie corner over the bar.
From fellow substitute Isaac Hayden's delivery, Mitrovic again went close, nodding the cross just past the post. Ritchie won a free kick from which Gayle went close, curling just off target.
And deep in added time, they equalised through Gouffran, who headed in a Yedlin cross from the right just seconds before Gayle completed his treble, firing in at the Leazes End to seal the win in the most dramatic fashion.
