Key Moments
The Red Devils' disappointing run of form continued at Elland Road, 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 woodwork for the in-form hosts before Brenden Aaronson applied a cool finish with a low, 62nd-minute shot.
With the goal, Amorim was under intense scrutiny after his side had failed to beat struggling Wolverhampton Wanderers at Old Trafford earlier in the week.
Nevertheless, against opponents who had two days less to recover, Cunha levelled three minutes after the opener when he latched onto a pass from introduced Joshua Zirkzee and slid a finish underneath rushing home goalkeeper Lucas Perri.
It was only the South American's fourth goal since his high-profile summer move and came at a time when Amorim is without four key attacking players due to a combination of fitness issues and national team commitments.
Not that Leeds to care about that given their own lack of preparation time.
The Swiss attacker forced a diving save out of visiting goalkeeper Senne Lammens with a spectacular overhead kick, and visiting heads were in hands when the Slovenian striker, who is finding it hard to make an impact, turned his teammate's cut-back wide from six yards out.
The goalscorer, who had a goal disallowed for offside much earlier in the contest, also fired against a post in the dying moments.
The draw extends Leeds' unbeaten run to seven matches, their best such streak since 2001, and takes them a significant margin clear of the drop zone.
For the Manchester United though, it was yet another opportunity to move into fourth spot, albeit only for a short time, wasted.
With the Africa Cup of Nations reaching its knockout phase, the United coach 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 exacerbated the impact the unavailability of other attackers is having on the team's options.
It's not that the missing pair are the answer to all of United'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 youngster Shea Lacey or the much more experienced Joshua Zirkzee, even though the Netherlands forward provided the assist for Cunha's equaliser almost instantly after his introduction as a replacement.
That leaves the Brazilian, who has now scored a small tally in recent outings, and others operating out of their natural positions, alongside Benjamin Sesko.
On this showing, the expensive frontman looks no better than previous underwhelming signings, lacking in belief and assurance in front of goal.
His failure to turn home the cross means it is now nine games without a goal for the young striker.
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 instances of expected fervor, but the hostile edge was not as pronounced as in other fixtures.
The home fans were more preoccupied about their side's struggle to get worthwhile possession in dangerous areas for any prolonged period.
The striker's header that clipped the post was an rare moment of threat in a congested first half.
It still felt as though Leeds were finding it tough when a mistake opened the door for Aaronson.
For a short moment, Leeds' ecstatic fans thought they were about to secure a rare win against Manchester United.
However, there is a wider context for Daniel Farke and Leeds to deal with.
When they endured a difficult run of form earlier in the season, many thought about the manager's position.
In contrast, he is now overseeing significant improvement.
The task is not done yet but gradually, Leeds are getting nearer to achieving their top-flight status.
The Yorkshire club have a Premier League trip to Newcastle in midweek, followed by the start of their FA Cup campaign next weekend. Manchester United face a league trip to Turf Moor before also entering their FA Cup journey with a home tie.
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