Dinamo were terrific in the first half, taking advantage of the heady atmosphere of the rather ramshackle National Stadium.
They were less so in the second 45 minutes and United looked to have settled into the unfamiliar positioning of Roy Keane at centre-back and John O'Shea in the holding role in midfield.
The first scare for United came in the fifth minute when Claudiu Niculescu curled a direct free-kick from 25 yards and Tim Howard managed to get a hand to the ball. For a moment it looked as though it would drop to Liviu Ciobotariu but he missed his chance and United heaved a sigh of relief.
That did not last long however as some uncertainty in midfield saw United opened up on their right flank. Niculescu carried the ball at speed into the box before firing a low cross to the far-post where Ionel Danciulescu prodded the ball back across the open goal - only for Fortune to slice it over the line as he tried to clear.
Paul Scholes tried to respond, meeting Fortune's cross with an instinctive header but straight at goalkeeper Vladimir Gaev.
Niculescu's speed was proving a concern and he again found himself free, running full pelt at Keane and drilling a shot that took a deflection off the Irishman into the sidenetting.
Dinamo continued to look threatening, especially on the break, and once more Keane was at full-stretch as Dorin Semeghin burst through. The full-back should perhaps have done better but a combination of Keane and Howard closed him out.
On the half-hour Dinamo were forced into a substitution with George Galamaz coming on for Ciobotariu, who had been crocked by a late Alan Smith challenge.
Seven minutes before the break though Giggs struck an excellent and valuable equaliser.
Dinamo lost possession on the halfway line, Scholes played the ball through first-time and Giggs raced full pelt towards goal, swerved around Gaev and finished with admirable coolness.
After the restart United appeared to be a different team - and a much-better one.
Scholes played a swift one-two with O'Shea before unleashing a thunderous strike from 25 yards which looked to have beaten Gaev but whistled inches past the post.
Smith, who had been booked after another scything challenge but was proving a painful thorn in the side if the Dinamo defence, went even closer when Gary Neville cut the ball back, striking a low shot against the inside of the post.
United continued to pressurise with Keane scuffing a half-chance before Howard pulled off a tremendous save to keep the scores level.
Ionut Balan's effort from a deep position on the right deep looked to be dipping at speed over Howard into the top corner but the American keeper somehow reached behind at full-stretch to turn the ball over the crossbar.
United were thankful to Howard again after Keane and Silvestre's misunderstanding led to another Dinamo breakaway. Ferguson's men still looked better going forward and Smith had a shot off-target after a good run by Fortune.
Miller, United's summer signing from Celtic, came on for the flagging Darren Fletcher after 67 minutes, and he was immediately involved with a cross for Scholes who attempted an outrageous overhead backheel that flew not too far off-target.
In the 70th minute Miller made an even more vital contribution. The 23-year-old picked up the ball in Dinamo's box and drilled the ball hard across goal, Alistar turning the ball into his own yet from close range.
Sir Alex Ferguson savoured a battling performance from his injury-hit side as Manchester United took a giant stride towards qualifying for the Champions League group stage.
Ferguson said: 'You have to be pleased to win away in Europe. After a bad start for us it was very pleasing to recover and go on and win this match.
'We knew they were a good team, there was some good forward play and we always had to be alive and alert.
'I am pleased with the team's positive performance and it gives us a great opportunity at Old Trafford.'
Ferguson had sprung a surprise in his formation by playing Roy Keane at centre-back to shore up United's fragile defence, with John O'Shea taking on the holding role in midfield.
The United manager added: 'Roy's contribution tonight was about his experience and to give confidence to those around him, that was the reason I played him there.
'I felt we still needed a positive presence in midfield and apart from that John O'Shea did very, very well on the ball for us.
'The last thing we wanted was to give away an early goal but we have been through it before and the most important thing was not to panic.'
Ferguson also played down the impact of a bomb scare at United's team hotel on the morning of the game.
He added: 'Apparently there was a phone call to the hotel saying there was a bomb but obviously it was a crank.
'The police searched all the rooms but it was nine o'clock so it was not a problem - the players just had a slightly earlier breakfast than usual.'
Dinamo coach Ioan Andone conceded that his side's chances in the second leg in a fortnight are now very slender.
Andone said: 'I am very sorry because we did not have a great game. Unfortunately after we went 1-0 up we did not know what to do.
'In the second half Manchester dominated the match. They created more opportunities, hit the post and we know it will be very difficult for us at Old Trafford.'
United striker Alan Smith was pleased with his European debut for the club.
'We have got a positive result,' Smith said. 'They started brightly, but we found our rhythm and started to play some good stuff - and could have got more goals.'
The former Leeds forward struck the post in the second half and he added on ITV 2: 'I was a bit unfortunate.
'But we are happy going back to Old Trafford with a 2-1 victory.
'We can't underestimate them in the second leg. But the lads stuck together, and it is a good win for us.'
Reports
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