The Australia international left the field after being struck on the head by a throw from the boundary.
Carberry (107) and Liam Dawson (87*) put on exactly 100 before Carberry was caught off the bowling of James Harris.
Harris, Toby Roland-Jones (3-87) and Tim Murtagh all struck in the final 10 overs as Hampshire closed on 315-7.
Hampshire lost opener Jimmy Adams early and captain James Vince, an England Test hopeful, followed for 25 when he was caught by Voges off Roland-Jones.
But Carberry was dropped at second slip shortly after passing his half century, with Murtagh the unlucky bowler, and this time made the most of his good fortune, having failed to convert a fifty into a three-figure score on 10 occasions in 2015.
Following his dismissal, Middlesex picked up the wickets of Adam Wheater, Ryan McLaren and Chris Wood to give themselves hope of containing Hampshire's total.
Should Voges be diagnosed with concussion, he will face a minimum of six days on the sidelines under the England & Wales Cricket Board's guidelines.
Middlesex managing director of cricket Angus Fraser told BBC Radio London:
"The feeling from the physio is that Adam has got concussion. It was a freak injury.
"It's sad for Adam and disappointing for us, as we are not going to have any more use of him in this game.
"He wasn't feeling very well in the dressing room so we decided to get him taken to hospital. He feels a lot better now and seems to be coming around.
"He wants to remain here and be part of what is taking place, even if he can't do anything on the field."
Michael Carberry's century gave Hampshire the edge against Middlesex, whose skipper Adam Voges was taken to hospital with suspected concussion.