The 2014 finalists moved through as the England Under-21 player showed his finishing power against the League One side.
The tie afforded Tigers chief Steve Bruce the opportunity to give fringe players an opportunity and he made 11 changes to the table-topping Championship side.
Sixteen-year-old debutant Josh Tymon became the youngest player in Hull’s history to play in the FA Cup, given his chance at left-back.
Central defender Michael Dawson also made a welcome return after nine weeks out with a knee problem.
Tom Pope led the Bury attack in the absence of illness victim Leon Clarke and in goal was Manchester City Republic of Ireland Under-21 loanee Ian Lawlor.
Much of the first half was played in a hailstorm and Bury threatened in the early stages but goalkeeper Eldin Jakupovic got a boot to a low drive from Barnsley-born former Oakwell striker Danny Rose.
Kelvin Etuhu limped off after fouling Sone Aluko and transfer window target Joe Riley came on at right-back with Tom Soares moving into midfield.
The Tigers struck in the 14th minute when Adama Diomande threaded the ball into the path of David Meyler racing down the middle. Lawlor did well to block the ensuing shot but Akpom hit the fierce rebound first-time into the unguarded net.
Hull settled after going ahead - Tymon making some fine runs - and Meyler had a header from a corner cleared off the line by Riley.
Bury still posed a threat and Dawson got a fine block in on Rose in a commanding return to action.
Bury made two changes soon after the break, and one of them, Ryan Lowe, soon turned in a drive from Mayor but was ruled offside.
Hull hit back and Chris Hussey, attempting to shepherd the ball out of play, tripped Sone Aluko as he nipped in to claim possession and a penalty was awarded. Akpom confidently sent Lawlor the wrong way to make it 2-0 in the 57th minute.
It was game over as Akpom was fed down the left channel by Aluko and curled home the third into the far corner as two defenders collided.
It was fitting reward for the 2,746 Hull fans in the 7.064 crowd.
Bury got a late consolation when Craig Jones controlled a cross from the left and scored at the far post to reward their never-say-die efforts in spite of lack of possession.
Hull manager Steve Bruce praised Akpom’s reaction as he hit an FA Cup hat-trick in City’s 3-1 win at Bury.
Bruce has overlooked Akpom since the Championship leaders’ defeat at Preston over Christmas and, according to the Tigers boss, the on-loan Arsenal striker “spat his dummy out last week”.
But the 20-year-old came back into the starting line-up at League One Bury and responded with a triple.
Bruce said: “I’m delighted for Chuba. He spat his dummy out last week and threw his toys out of the pram.
“He’s a young player with lots of ability and I’m sure it’ll do him the world of good. It’s good to see him get a hat-trick and I’m sure he’ll remember that for a while.
“I used the squad, like I always do, and if I can call on players like that then it bodes well. It was difficult for us because of the conditions but I was pleased with the way we went about it.
“To be fair to Bury, they were spirited and aggressive, and they made it a proper cup tie.
“In the end we got the breaks and that little bit of quality saw us through.”