By then they will know whether fourth is possible and they will need Benetton, who haven’t won all season, to deny the Warriors a bonus-point win at Scotstoun on Saturday.
Director of rugby Dean Ryan had called on his side to stay alive into their final weekend and they did that thanks to impressive physicality in defence.
They did that despite the loss through injury of Ollie Griffiths and Jamie Robert in the opening exchanges and a 17-9 half-time deficit.
The Dragons fought back thanks to tries by full-back Jordan Williams and lock Matthew Screech then stood firm impressively at the death.
That could turn out to be crucial – by making it a pointless afternoon for Glasgow they have heaped the pressure on them to win with a bonus against the Italians.
It was an impressive win with the pack putting in a monumental shift; fly-half Sam Davies took the official man of the match honours but it was a success earned up front.
Captain Harri Keddie and fellow back rower Ross Moriarty stood out with some huge hits along with front rowers Richard Hibbard and Lloyd Fairbrother, who also enjoyed scrum dominance.
Even if the Dragons don’t sneak into the Champions Cup, this was a win that gives hope they are on the right track.
The Dragons suffered a pre-match blow in their final drill before returning to their changing room when Josh Lewis went down in agony after twisting his right leg.
The full-back was helped from the field – he would later return to the pitchside with his ankle in a moonboot – with Jordan Williams given a very late recall to the starting line-up.
It was a manic opening after Glasgow kicked a penalty to the 22 and hammered away only to turn the ball over on the line and then be forced into desperate defence to stop fit-again wing Ashton Hewitt going the distance.
The Dragons struck first when Davies punished a breakdown offence from the tee in the seventh minute and they then defended strongly to protect the 3-0 lead.
However, they suffered a blow in the 11th minute when influential back rower Griffiths was forced off after yet more injury misfortune; he tried to play on after having his bicep strapped but it was no good.
Ben Fry came on wanting to make an impact and he did just that, although in a negative way after being shown yellow for a dangerous, reckless, daft no-arms chop tackle.
Glasgow immediately made the hosts pay with a neat grubber kick to earn a close-range lineout that Scotland hooker Fraser Brown, on his first outing since an Autumn Nations Cup injury, guided over the line.
The Dragons suffered another injury blow when Roberts was forced off by a knee problem with Wales squad centre Nick Tompkins replacing him and soon it was 14 versus 14.
A searing line-break by Hewitt was ended by a high tackle by Adam Hastings and he was shown yellow with Davies knocking over the penalty to make it 7-6 to the visitors after a quarter of the game.
The Dragons thought they were in front when, soon after Fry had returned, Gonzalo Bertranou darted over from a quick tap.
However, referee Adam Jones deemed after watching the footage that the Argentina scrum-half was initially short of the line and that he then lost control of the ball.
It was an extremely harsh call but the Dragons stayed compose and eventually inched in front through Davies’ left boot.
However, they were chasing again after the Warriors scored their second after 36 minutes thanks to a combination of superb finishing and shoddy defence.
Poor lineout ball fell to wing Rufus McLean and he jinked past forwards, did Hewitt with his fast feet and then had too much pace for the covering Jordan Williams.
Hastings twisted the knife from the touchline and then struck with the final kick of the half to make it 17-9 at the break.
The Dragons were on the front foot from the restart as they hunted a comeback win and they were once again denied a try by video footage when Ross Moriarty held onto defenders before Nick Tompkins crossed from a driving lineout.
They went back to penalty advantage – Ryan Wilson was sin-binned – but turned the ball over after a brain-dead decision to go for the quick tap rather than return to the corner.
The Dragons kept the pressure on, Glasgow kept infringing but Mr Jones opted not to reduce them to 13.
The clock was not the hosts’ friend in the hunt for four tries but winning was first job and Davies got them back to within striking distance with a penalty to make it 17-12 after 51 minutes.
They then showed more strong defence after Warriors pressure and it paid off in the 63rd minute when Aneurin Owen’s neat pass put Jordan Williams clear and he finished well, Davies’ conversion earning a 19-17 lead.
It got better with the forwards hammering away at the line before Matthew Screech powered over.
Davies made it 26-17 and the Dragons had 13 minutes to take the spoils, and potentially score two more tries to turn a fine win into a sensational one.
Yet they instead had to concentrate on earning the success with some superb defence in their 22.
Rhodri Williams was shown yellow for a professional foul with four minutes left but the Dragons stood firm against the direct Warriors attack.
Glasgow mystifyingly kicked a penalty to the corner and failed to barge their way over. The Dragons have shown that they are no longer a soft touch.
Dragons: J Williams; J Holmes, A Owen, J Roberts (N Tompkins 16), A Hewitt; S Davies, G Bertranou (R Williams 61); B Harris (J Reynolds 76), R Hibbard, L Fairbrother, J Davies (B Carter 61), M Screech, H Keddie (captain), O Griffiths (B Fry 10), R Moriarty.
Scorers: tries – J Williams, M Screech; conversions – S Davies (2); penalties – S Davies (4) Glasgow: A Hastings; R McLean, N Grigg, R Fergusson, L Jones; R Thompson, S Kennedy; A Seiuli, F Brown, E Pieretto, R Gray, K McDonald, R Harley, T Gordon, R Wilson (captain).
Replacements: G Stewart, D Evans, D Rae, L Nakarawa, F Lokotui, J Dobie, P Horne, O Smith.
Scorers: tries – F Brown, R McLean; conversions – A Hastings (2); penalty – A Hastings Referee: Daniel Jones (WRU)
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