Match Summary
The Red Devils' poor run of form continued at Leeds' fortress, but head coach the Portuguese tactician had his Brazilian forward to thank for the team not suffering even more disappointment against Leeds.
Dominic Calvert-Lewin headed a first-period effort against the post for the confident hosts before the American midfielder finished superbly with a driven, 62nd-minute shot.
With the goal, Amorim was in the firing line after his side had been held by struggling Wolves at Old Trafford earlier in the week.
Nevertheless, against a team who had less time to recover, Cunha restored parity three minutes later when he latched onto a through ball from introduced the Dutchman and guided a finish underneath advancing Leeds goalkeeper Lucas Perri.
It was only the South American's fourth goal since his big-money summer move and came at a time when the manager is without four key attacking players due to a combination of injury and national team commitments.
Not that the hosts to care about that given their own lack of preparation time.
The Swiss attacker forced a full length save out of visiting goalkeeper Senne Lammens with a spectacular overhead kick, and United heads were in hands when the Slovenian striker, who is struggling to make an impact, turned Zirkzee's cut-back wide from six yards out.
The goalscorer, who had a goal disallowed for offside much earlier in the match, also blasted against a post late on.
The point prolongs the home side's unbeaten run to seven games, their best such streak since 2001, and takes them eight points clear of the drop zone.
For the visitors though, it was yet another chance to move into the Champions League places, albeit only for a short time, wasted.
With the Africa Cup of Nations reaching its knockout phase, Amorim could be forgiven for hoping the involved countries make a pretty quick exit.
Current injuries to skipper Bruno Fernandes and Three Lions star Mason Mount only heightened the impact the unavailability of other forwards is having on the team's squad.
It's not that the absent players are the answer to all of the club's problems. After all, they were part of the side that experienced poor results earlier in the season.
But without them, the boss has limited alternatives in attack.
He evidently does not have full faith in teenager Shea Lacey or the much more experienced Zirkzee, even though the Netherlands forward delivered the assist for Cunha's equaliser almost immediately after his introduction as a substitute.
That leaves Cunha, who has now scored three goals in five games, and others playing out of their preferred positions, alongside Benjamin Sesko.
On this evidence, the expensive frontman looks similar to previous struggling signings, lacking in confidence and conviction in front of goal.
Sesko's failure to turn home Zirkzee's cross means it is now a long run without a goal for the new signing.
After thrilling victories in their last few home games, the arrival of their fiercest rivals to home was a comparatively subdued affair.
True, there were moments of expected atmosphere, but the intimidating edge was not as pronounced as in previous encounters.
Leeds' fans were more concerned about their side's inability to get sustained possession in advanced areas for any prolonged period.
The Calvert-Lewin header that struck the post was an rare moment of quality in a tight first half.
It still felt as though the hosts were finding it tough when a defensive error opened the door for the opener.
Ever so briefly, Leeds' ecstatic fans thought they were about to secure a rare win against their old foes.
Nevertheless, there is a wider context for Daniel Farke and Leeds to deal with.
When they endured a difficult run of form earlier in the campaign, many thought about the manager's position.
In contrast, he is now masterminding clear progress.
The task is not done yet but gradually, the team are getting closer to achieving their top-flight status.
Leeds have a top-flight trip to St James' Park in midweek, followed by the start of their FA Cup campaign next weekend. Manchester United face a league trip to Burnley before also beginning their cup journey with a home tie.