She was performing her sell-out tour at the SSE Hydro on Friday night when a chain fell from the production rigging onto a woman in the audience.
She was taken to hospital "as a precaution", a spokesman for the venue said.
Adele later tweeted: "I'm so sorry to hear that someone got hurt at my show tonight.
"It's being investigated to ensure it won't happen again. X"
A spokesman for the venue said: "The lady was attended to by our own medical team and then taken to hospital as a precaution.
"The safety of every member of our audience is paramount and we are currently working with the promoter's production team to establish why this happened and to ensure it does not happen again."
Stuart Duffy, who was at the show, said the lights came up and other spectators were moved.
He posted a picture on Twitter of the rigging that appeared to have malfunctioned.
Adele will perform a second show at the 13,000-capacity arena on Saturday.
Her album 25 stayed at the number one spot for the 12th consecutive week, denying rock band James - who had been leading the chart earlier in the week - the top spot.
Pop star Adele has apologised after a fan was injured by a falling chain at a concert in Glasgow.