The match took place at ADO Den Haag's ground on an artificial surface after the collapse of part of a stand at AZ's own stadium meant the Dutch outfit were unable to officially host the Red Devils.
Both teams had half-chances during the 90 minutes of football, but neither could find the back of the net, leaving United top of the section with four points from their two Group L fixtures and AZ third with two.
Solskjaer made eight changes from the team that started against Arsenal, and there was a full debut for academy product Brandon Williams at left-back, while Mason Greenwood and Angel Gomes also came into the XI.
It did not take long for United to gain control of the possession, but AZ had the first half-chance of the match as a brilliant low cross from Owen Wijndal had to be cleared by Marcos Rojo in front of David de Gea.
AZ actually had the ball in the back of the net from the resulting corner through Boadu, but the offside flag was raised against the number nine, much to the relief of the English giants.
The match settled down around the 30-minute mark as United enjoyed more of the ball once again, but the 20-time English champions were struggling to make things happen in the final third.
Diogo Dalot and Williams were finding space in forward positions, although the full-backs were struggling for targets in the box. United did have a half-chance in the 30th minute when Daniel James found Greenwood in a dangerous position, but the teenager saw his strike blocked by former Aston Villa defender Ron Vlaar.
AZ had an opportunity of their own in the 32nd minute when Stengs found Boadu inside the United box, but the forward could only head into the arms of De Gea from close range.
Gomes then failed to make proper connection on a dangerous cross from Dalot down the other end as the visitors threatened to take the lead 10 minutes before the interval. As it transpired, though, the two teams headed down the tunnel at half time with Solskjaer having plenty to ponder ahead of the second period.
AZ were causing problems in the wide areas in the early stages of the second half with Idrissi and Stengs both enjoying space, but United continued to defend their lines under pressure. The hosts had a shout for a penalty when the ball hit the hand of Williams, although the referee was not interested.
Yukinari Sugawara then fired over the away side's crossbar just before the hour as AZ continued to enjoy positive moments in the final third of the field.
Marcus Rashford came off the United bench in the 63rd minute, but the pattern of the match did not really change. AZ had another shot in the 71st minute through Wijndal, although it was comfortable for De Gea.
Idrissi then struck a powerful effort wide of the post before United made their second change - Jesse Lingard entering the field in place of Greenwood.
United had a huge shout for a penalty in the 80th minute when Rashford went down inside the box under a challenge from Stijn Wuytens, but neither the referee nor the linesman signalled for a spot kick.
Lingard then almost found the top corner with a powerful drive in the 84th minute after smart work from Scott McTominay, who was the third player off the away side's bench. There was to be no late drama in the Netherlands in terms of goals, but United finished with 10 men as Lingard limped off in the final exchanges.
AZ will be looking for their fourth straight league win when they travel to Willem II on Sunday, while United are next in Premier League action away to Newcastle United on the same afternoon.
AZ (4-3-3): Bizot; Wijndal, Vlaar, Wuytens, Sugawara; Koopmeiners, De Wit (Hatzidiakos 87'), Midtsjo; Idrissi, Boadu, Stengs
MANCHESTER UNITED (4-2-3-1): De Gea; Dalot, Lindelof, Rojo, Williams; Fred, Matic; James (Rashford 63'), Gomes, Mata (McTominay 83'); Greenwood (Lingard 77')