Top-scoring out-half Gary Cribb dinked a kick through to the corner where winger Shay O'Donovan beat Jonny Cullen to the ball for an unconverted try.
'Hinch, who were missing some key players, opened their account in the 11th-minute when former Ireland Under-19 international Harry McAleese kicked a penalty.
But Clon tightened their grip on the game soon after when the home side's Davy Harris was intercepted in midfield and scrum half Joe Crawley had the pace to run in the touchdown for the Corkmen from the halfway line.
David Lombard converted and the Clon captain also fired successive penalties - after 28 and 37 minutes - through the posts to give the visitors an 18-3 half-time lead.
'Hinch, turning around with the elements at their backs, made a good start to the second half with Michael Graham muscling his way through in the 51st-minute and sliding over for a well-taken try.
McAleese missed the conversion but 'Hinch were turning things around, in terms of possession and territory, and had they shown more composure in the backs, they may have nabbed more tries.
The County Down side were stung in the 68th-minute when Clon burst out of their 22 on the counter, stringing together some excellent passes before Cribb crashed over the whitewash for an unconverted try.
That put the game beyond 'Hinch but they did put Cullen over for a late consolation try, which McAleese converted.
BELFAST HARLEQUINS 38 BARNHALL 3, Deramore Park
Scorers: Belfast Harlequins: Tries: Stuart McComb, Shane Stewart, David Creighton, Andrew Gillespie, Paul McKenzie, Colin Atkinson; Cons: Darren Cave 4
Barnhall: Pen: Jamie Kavanagh
After seven seasons in Division Two, Kildare club Barnhall have been relegated with Saturday's heavy defeat to Belfast Harlequins sending them down.
This was very meaningful game for both teams. Harlequins were hoping to grab a bonus point win and give themselves a last chance at gaining promotion straight back to the top flight.
Other results did not go their way however, and in the final table, third-placed 'Quins (51 points) finished just behind Young Munster (52) and Buccaneers (53). Nonetheless, Andy Ward's men will still feature in the play-offs.
For Barnhall, this was a highly-charged and emotional afternoon which ended in tears for a lot of their players and supporters.
Summing it up afterwards, Barnhall President Jack Looby said: "On behalf of all of the Barnhall supporters, we thank all of the boys in blue for the effort, dedication and commitment they have given to Barnhall in this difficult season and for all of our years in Division Two."
The visitors did get off to a good start with Jamie Kavanagh booting a third-minute penalty, but they missed out on some early scoring chances.
Kavanagh missed a subsequent kick and a quickly taken tap penalty by Paul Burke came to nothing. A poor final pass also eluded Romain Gibert as the Blues Bulls threatened to score a try.
'Quins managed that feat in the 21st-minute. Stuart McComb burst his way through for a try converted by Ireland 'A' international Darren Cave. That left the home side leading 7-3 at the break.