The Clarets took the lead after nine minutes when Tom Wraight stroked home before former City player Jordan Parkes levelled for the hosts moments later.
Billy Clifford gave the Tudors a second half lead but two goals in two minutes from Anthony Church and Rob Swaine put City ahead 2-3.
Parkes’ inch-perfect freekick levelled the scores once again but goals from Rhys Murphy and Chris Whelpdale secured all three points from an end-to-end fixture.
City boss Rod Stringer made three changes to the side that beat Bath City 1-0: Scott Fenwick, Taylor Miles and Matt Young made the starting eleven at the expense of Jazzi Barnum-Bobb, Jai Reason and Chris Whelpdale.
The Clarets started brightly as they utilised Wraight’s ability on the wing. The wide-man almost carved open an opening goal with his drilled low cross but Laurie Walker claimed the delivery. A long throw to Swaine fell at Church’s feet but the captain’s tame effort went wide.
However the effort finally reaped rewards as Wraight opened the scoring on nine minutes. Matt Young, playing at full-back, drove into acres of space on the right-wing, his cross picked out Taylor Miles inside the area but the midfielder had his shot blocked. The deflected effort fell to Wraight who controlled under pressure and kept his composure to stab home from close range.
City claimed a penalty when Swaine fell to the ground during a corner but the weak appeals were waved away by referee Mr Cartmel.
Moments after taking the lead the hosts were level. Joe Anderson failed to clear the danger on the left-flank, Karl Oliyide won the ball and delivered a low cross into a crowded box where Parkes was able to sweep a low shot past McDonald.
The hosts could have taken the lead however Charlie Sheringham was unable to keep his header on-target, while Clifford could only muster a low shot.
Church’s chipped through-ball allowed Wraight to race through-on-goal against Walker, but the midfielder’s low shot was saved by Walker’s legs as an action packed first half reached the half-hour mark.
City looked to take advantage as Oliyide lay on the floor injured when Anderson found Murphy inside the area, but the striker slipped as he attempted to turn under pressure and the danger was cleared.
Swaine’s volley from a deep Anderson freekick forced a strong save from Walker, before the tall defender was unable to connect with Miles’ delivery as he waited at the far-post just before the break.
With the hosts looking to exploiting vulnerabilities down the left wing, Watt almost gave the Tudors the lead but Max Porter cleared it off the line.
Although the hosts were soon ahead under controversial circumstances. Swaine appeared to be fouled inside the area but, as the referee waited to make a decision, the ball fell to Clifford and the midfielder fired home from close range and the goal stood.
The Tudors could have increased their lead when Parkes slid Sheringham through-on-goal, although the son of former England international Teddy shot straight into McDonald’s palms.
With City still searching for a goal, Stringer replaced Porter with Jai Reason on 62 minutes.
Murphy could have restored parity when he latched onto a deflected pass but, after turning the ball onto his right foot, he was forced to settle for a corner as James Kaloczi tracked back to defender.
But the hosts could not defend the set-piece as it bounced inside the six-yard box and Church leant forward to nod home.
Two minutes later City found themselves ahead. Defender Swaine controlled excellently inside the area and rammed home his first goal for the Clarets.
Tom Hamblin’s attempted back-pass to Walker almost gave City a fourth, but Scott Fenwick’s effort was straight at the Tudors’ shot-stopper.
Parkes levelled the scores for a third time as his curling freekick struck the post and cannoned past McDonald.
Stringer made his second change with eleven minutes remaining, swapping Jonny Giles for Scott Fenwick. And the tricky winger was involved immediately as two minutes later City took the lead.
Giles’ through-ball split the Hemel defence and as Murphy ran through-on-goal, the City striker calmly placed a shot over Walker and into the net much to the jubilation of the Clarets supporters.
Murphy almost scored again when he tried lobbing Walker for a second time, but the goalkeeper was unmoved and claimed the striker’s shot.
But substitute Whelpdale wrapped up proceedings in the final minute. The experienced midfielder turned cleverly inside the area, shifted the ball onto his right foot and sent a deft lob over Walker to nestle into the bottom corner.
Wraight almost added a sixth for City but the winger’s low shot was saved by Walker as City took all three points from another entertaining fixture