Ipswich, who began the day in the relegation zone, fought hard and managed to secure the equaliser they had worked for with a goal from Omari Hutchinson just before the first half ended.
Neither side could find a winner in the second period despite chances at both ends, with the result leaving both managers unsatisfied.
United are a lowly 12th in the Premier League, six points behind fourth-placed Arsenal, while Ipswich are still in the drop zone.
Amorim, who cut an animated figure on the touchline, said his players were guilty of over-thinking during the game against Kieran McKenna's men.
"I think my players were thinking too much," he told . "We lost some balls without pressure. But they really, really tried and they want this."
He added: "This week I spoke more (to the media) than I did in four years at Sporting. I just want to work with my players. Nothing more."
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'Great again'
Before kick-off two United fans held up a home-made sign saying: "Good luck Ruben Amorim, Make our team great again."
The Portuguese, who replaced the sacked Erik ten Hag, made three changes from the final line-up named by United's interim boss Ruud van Nistelrooy before the international break.
And he changed his formation to play three at the back as widely expected, with England forward Rashford operating as a lone central striker ahead of captain Bruno Fernandes and Alejandro Garnacho.
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United were ahead within two minutes when Diallo surged down the right, shrugging off the attentions of Leif Davis and Jens Cajuste to whip in a cross that Rashford touched home from close range.
Amorim, wearing a black coat and dark trousers, remained impassive in his seat as United's players celebrated in the corner.
Christian Eriksen flashed wide from just outside the box in the seventh minute after more incisive work from Diallo.
But Ipswich responded well, creating openings of their own and looking the more dangerous team as the half wore on.
The home side should have been level in the 40th minute but Andre Onana somehow kept out a close-range effort from Liam Delap with a strong right hand before Cajuste flashed over the bar.
Ipswich got the equaliser they deserved just before half-time when Hutchinson hit a stinging shot that skimmed off the head of Noussair Mazraoui and beyond the diving Onana.
The watching Ed Sheeran, a boyhood Ipswich fan, celebrated wildly in the stands among the 30,000-strong crowd.
Garnacho had a chance to put United back ahead in the opening moments of the second half but Arijanet Muric got down to his left to keep out the shot.
Onana saved well with his foot from a Delap flick from close range before Cajuste produced a perfectly timed challenge at the other end to cut out a Garnacho pass to Rashford after a swift break.
Amorim threw on Luke Shaw and Manuel Ugarte for Jonny Evans and Casemiro about 10 minutes after the restart but United continued to labour against a team with just one Premier League win all season.
He later introduced forwards Rasmus Hojlund and Joshua Zirkzee to add fresh legs to his attack.
Fernandes hit a free-kick just wide of the post from a central area and Onana saved from Conor Chaplin at the other end but neither side found the quality to earn the win.
Amorim is United's sixth permanent appointment since Alex Ferguson retired in 2013 after leading to the team to a 20th English title.
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At 39, he is the youngest United boss since Wilf McGuinness followed Matt Busby in 1969.
(This story has not been edited by staff and is auto-generated from a syndicated feed.)