Handscomb (95) shared a partnership of 177 for the sixth wicket with Rehan Ahmed (90) as the hosts put on 407 after being in trouble at 5-168.
The Victorian is in fine early-season form in the UK, sitting third in the second-division top run-scorers list.
But despite playing all four Tests on Australia's recent tour of India he was overlooked for their World Test Championship final and initial Ashes squad.
Handscomb was last man out for Leicestershire on Friday, throwing back his head in frustration as umpire Neil Bainton upheld Queenslander Michael Neser's (2-98) call for a caught behind.
In response, the visitors closed the day still 94 runs short of avoiding the follow-on at 5-164.
Test star Marnus Labuschagne joined his fellow Australian Hadscomb in the runs with a neat 64 for Glamorgan before falling to the bowling of Ed Barnes.
Barnes, who had been pulled for six by Labuschagne a couple of overs earlier, surprised him with another shortish delivery that found the edge, Wiaan Mulder at second slip grabbing the catch at the second attempt.
It was part of a mini-collapse for the visitors who looked well-placed at 1-123 before the late flurry of wickets.
Sam Northeast was not out on 16 after being dropped on nine and is Glamorgan's main hope to stay in the game.
Earlier, Handscomb batted briskly with Ahmed, at one stage adding 72 in the 13.2 overs before the new ball.
In London, South Australian quick Wes Agar enjoyed a strong outing in his first match at Lord's, snaring four wickets against Middlesex including former Test openers Sam Robson and Mark Stoneman.
