Murray overcame the dangerous Alexander Bublik in his opening match, but he has found it difficult to string a run of wins together since his hip operation three years ago - and it was no different against the 21-year-old, world number nine.
The three-time grand slam winner did not manage to force a single break point in their only previous meeting, a straight-sets defeat at the 2020 US Open.
He righted that wrong by breaking in the fifth game of the first set, but Auger Aliassime had already won the previous four games.
The Scot, who won this tournament in 2009, clawed a break back early in the second but smashed his racket in frustration as he immediately dropped serve again to trail 3-2.
There was a bizarre moment on match point when Murray's cap flew off, causing him to stop playing as Auger Aliassime hit a winner.
He was afforded a let, but Auger Aliassime duly went on to convert a second match point for a 6-3 6-4 win and a meeting with Great Britain's Cameron Norrie in Friday's quarter-final.
Norrie won his second game of 2022 with a straight sets win over Karen Khachanov.
Speaking to Amazon Prime after the win, Norrie said: "Definitely wasn't easy - every time I play Karen we have battles so I knew what I was in for today.
"I think both of us donated our serves at times and both of us played great, so it was down to who could maintain that level and I managed to sneak through in that tie-break. I actually thought I returned really well today.
"With Karen you want to be the one dictating play so I wanted to use my forehand line a little bit more than usual today. I think the one being able to use their backhand more and being a bit more aggressive would have more success.
"Everyone can play well here at this tournament so it's going to get even tougher after my next match and hopefully someone can watch me play in my next match! No fans for the last two matches so hopefully I can show the Dutch crowd a bit of my game."