The Irish visitors were on top for much of the contest as Warriors failed to find top gear.
But McBeth squeezed over with five minutes left to settle a sizzling clash at Scotstoun.
Head coach Franco Smith said: "Fortunately we got it done, even if we made it hard for ourselves.
"We were our own enemies for most of it.
"We missed touch three times from penalties, lost the gain line most of the time and couldn't get our attack shape.
"But credit to Munster, they are a well-coached team. I'm happy to win." Warriors 2015 Pro12 champions were given a warm welcome beforehand on the anniversary of that triumph.
It was a big night, too, for Sione Vailanu as he made his first appearance since December 2023.
Glasgow soon found themselves with a man advantage after lock Fineen Wycherley slammed Alex Samuel to the ground.
Warriors tried to make the most of his 10 minutes in the sin bin but Johnny Matthews over-threw a lineout and the chance was gone. The away side were still a man down when they moved in front as Tony Butler's clever crossfield kick was caught by Ben O'Connor who fed John Hodnett to score.
Then they battered away at the Glasgow line before Paddy Patterson moved it wide and Sean O'Brien stretched over.
image [https://cdn.magzter.com/1583517504/1742610876/articles/4ZcCM87ma1742634297161/2126117216.jpg]
Glasgow hit back with a stunning team effort.
Ollie Smith and Stafford McDowall were both involved before George Horne finished under the posts. That took him joint top as Warriors' all-tim...