Matt Coles and Darren Stevens removed Mark Wallace and Jacques Rudolph in the first two overs.
But Will Bragg (33 no) and Chris Cooke (42 no) played positively in response, although Cooke was dropped on 23.
Kent keeper Adam Rouse, a late call-up for Sam Billings who suffered a hand injury in practice, needed treatment for a dislocated finger.
Glamorgan rested in-form South Africa batsman Colin Ingram, who missed the first six weeks of the season because of a knee problem.
Adam Rouse told BBC Radio Kent:
"I was playing club cricket in Hampshire when I got the call, Sam had been hit on the thumb.
"I wasn't expecting this game but Sam's got the T20 (England) stuff coming up and that's important, he wants to make sure he's right for that.
"The ball was wobbling around a bit and I got one on the end of the finger, [physio] Dan [George] popped it in and chucked me a few pain-killers.
"We got a couple of early wickets but Cooke's struck them nicely, it's definitely the conditions for swing bowling."
Glamorgan recovered from a poor start to reach 93-2 before rain halted play before lunch against Kent in Cardiff.