The Quins scored three tries and conceded four in a thrilling tie which was effectively won by a memorable score from England wing Ben Cohen.
It means Quins need a minor miracle to make the quarter-finals, but Kingston insisted his side still have a point or two to prove at Edgeley Park.
“It was win or bust,” Kingston admitted. “We gave away so many soft tries in the first half that we had to really go for it in the second.
“They had four chances and they scored four tries. It’s been a massively disappointing Heineken Cup campaign, but we should warn Sale that we won’t travel all the way up there next weekend without being competitive.”
Elsewhere, Leicester staged a late fightback but couldn’t prevent a potentially-damaging 40-30 defeat to Clermont Auvergne.
The French side raced into a strong early lead amid snow showers, but despite the late flurry which led to tries for Anthony Allen, Toby Flood and Jeremy Staunton, the Tigers were always second best.
Bath kept their hopes alive by holding on for a 16-9 victory over Edinburgh for their first win of the campaign.
Tries from hooker Pieter Dixon and Michael Stephenson proved enough to cancel out Chris Paterson’s three penalties.