The Portuguese winger was moved up front to help Fulham cope with the absence of Saha, who was sold to Old Trafford a month ago and made his first return to West London.
Saha was booed from the start by Fulham fans angry at the manner of his departure, but he thought he had got the last laugh after blasting United ahead after only 14 minutes.
But Fulham staged a fine second-half comeback to equalise through Boa Morte after 64 minutes and leave United even further behind in the race for the Premiership title.
United manager Sir Alex Ferguson managed to make headlines even before the game kicked off, thanks to an astonishing team selection.
He dropped Ruud van Nistelrooy, Ryan Giggs and goalkeeper Tim Howard, three of United's most impressive performers this season, to the bench in a ruthless change of tactics.
Ferguson, clearly angered by his team's midweek Champions League defeat in Porto and the previous weekend's tepid draw against lowly Leeds, wielded the axe with venomous effect.
With Gary Neville suspended and Mikael Silvestre still injured he fielded yet another new-look defence - with Roy Keane taking his place at centre-half and there were recalls for the likes of Diego Forlan, Cristiano Ronaldo and Darren Fletcher.
The only good news for United fans was that Saha was fit to play against his old club.
As for Fulham, they were without the injured Lee Clark and Jerome Bonnissel and missing suspended Carlos Bocanegra and handed young full-back Adam Green a rare chance at left-back.
The biggest absentee for the home side, however, was manager Chris Coleman, still stuck in hospital recovering from an infection.
He picked the side from his hospital bed with the help of assistant Steve Kean, but it was Fulham who were on the back foot right from the start.
Saha could have scored after only 30 seconds when, straight from the kick-off, he was put through by Forlan, only to whip his cross just inches wideRonaldo then served notice of United's attacking intentions by hitting a shot wide and another straight at Edwin Van der Sar.
And the goal came after only 14 minutes when Roy Carroll bowled out a glorious throw to Ronaldo on the left.
He passed immediately inside to Saha, who brushed off his marker to power into the box and lash a right-foot shot high into the middle of the net and over Van der Sar.
It was fairytale stuff for the French striker, who was booed throughout by Fulham's fans, having ditched the club to move to United in a £12.8million deal four weeks ago.
Fulham almost equalised immediately when Saha's replacement Brian McBride shot wide from a Bobby Petta cross.
But from that point on it was the visitors who ran the game, with Fulham guilty of giving the ball away too often.
The nearest they came to equalising in the first half was when Sylvain Legwinski's header from a Sean Davis corner bounced off the bar. And then, just before the break, McBride headed a good opportunity wide from a Petta cross.
United were worthy of their half-time lead and should have gone further ahead when Ronaldo produced a glorious piece of magic just after the break.
The Potuguese winger produced a stunning turn on the left wing, cut inside and curled shot narrowly wide.
Forlan then headed straight at keeper Van der Sar when well placed, and United were made to pay after 64 minutes when Boa Morte raced on to a throughball, brushed aside a weak challenge from Wes Brown and drilled a low shot underneath Carroll's arm and into the net for 1-1.
United brought on van Nistelrooy and Giggs in a bid to save their title challenge and twice came agonisingly close to victory.
Saha was denied what looked a certain penalty when he was put clean through on goal, only to be blocked by keeper Van der Sar. But when the Frenchman fell to the floor, referee Alan Wiley waved play on.
Van Nistelrooy then went even closer when he raced on to a Giggs throughball, but Van der Sar just managed to get a hand to the ball as he tried to dink it over himBut Fulham held on for a memorable point and left United wondering just where their season is going.
Manchester United manager Sir Alex Ferguson was furious with referee Alan Wiley for not giving a penalty that could have won his side victory.
He thought Saha was brought down by van der Sar in the second half and said: "You can see it yourself, I don't have to say anything. If these referees aren't going to do their job they shouldn't be there.
"If this is meant to be the dawn of professional referees I'm really not sure about that.
"It sometimes happens when you are on a bad run that things don't go for you. Maybe in the future it will start to go our way.
"But that decision could cost us a lot, everybody knows that. It's time the match officials earned their money.
"However football is a funny game and the most important thing is we know we are capable of winning matches. There's every chance we can go on a run of victories and that's what we need to do."
The only consolation for Ferguson was the form of Saha, who shone against his former club.
He said: "Louis Saha has been fantastic. The more I see of him the more I am happy. He scored a marvellous goal, he accelerated between players and hit a really great shot."
Man of the match: Luis Boa Morte - The Portuguese star was moved up front to try and replace his old pal Louis Saha and responded with a bustling all-action performance and a crucial goal.