Hutchins scored five goals in the competition that season and as the former winger recalls there is a striking similarity between the club's latest giant-killing run and the one four decades ago.
The class of 2015 followed up last month's epic fourth-round win at Chelsea with victory over Sunderland and have been drawn to play manager Phil Parkinson's former club Reading at home for a place in the semi-finals.
Parkinson and his players admitted after Sunday's triumph that the controversial decision by television bosses not to broadcast their fifth-round tie live had galvanised the dressing room, and Hutchins remembers that same siege mentality had steeled Bradford back in '76
"We were in the old Fourth Division and drew Norwich away in the fifth round," Hutchins told Press Association Sport.
"They were going well in the First Division at the time
They beat Arsenal in their previous game but their manager John Bond really wound us up.
"The tie was postponed on the Saturday and rescheduled the following midweek because several of our players were ill and we only had a small squad.
"Well, John Bond said if we couldn't put a team out then we shouldn't be in the cup and his comments got us all going.
"It gave us that extra edge and the same can be said of the team on Sunday.
"The television people can say what they like about why they didn't cover the game live but, like us 39 years ago, the players got extra motivation."
Hutchins struck the game's first goal against Norwich at Carrow Road and Bradford went on to win 2-1 to secure a home quarter-final tie against Southampton.
But City's fairytale run had a controversial ending
Southampton's Peter Osgood flicked the ball up direct from a free-kick for Jim McCalliog to volley home the game's only goal - later deemed unlawful - and the Saints went on to lift the Cup.
"We were probably the better team that day," said Hutchins
"(Southampton manager) Lawrie McMenemy said after they won the final that the game against us was the hardest they'd had
We didn't deserve to lose.
"I was in the wall for the free-kick and McCalliog volleyed it over us and into the net
We didn't realise then that the goal shouldn't have stood.
"I can't remember when we became aware of that
Probably when they highlighted it on Match of the Day.
"Somebody picked up on it
The ball must roll a full circumference before someone else can touch it
But that's life, it's history now
At least we lost to the eventual winners."
Hutchins, who left Valley Parade in 1981 and went on to work for Johnstone's Paints, insists that Cup run was the springboard for promotion the following season and feels Parkinson's side could follow suit.
"Without doubt it led to us going up the year after," he said
"It was more or less the same team and we went unbeaten at home in the league.
"Parkinson has hit on the right blend in his team and this Cup run could be their catalyst."
Parkinson steered Bradford to promotion from Sky Bet League Two in 2013, little over two months after the club had famously become the first from the bottom tier to reach a major cup final at Wembley in the Capital One Cup.
"That experience has stood them in good stead for what they've done this year," Hutchins said.
"They don't seem overawed by anyone now and Parkinson deserves a lot of credit.
"You can't take on the top teams that they have beaten without proper preparation."
Now retired, Hutchins still lives in Bradford but while closely following the club's latest exploits admits to being "a hopeless watcher".
"I'm a bag of nerves," he said
"Even when they're on the telly I can't really watch.
"I'm usually behind the settee and I'll probably be there for the Reading game."
Source : PA