Paul Turner's men currently lie in sixth position in the standings, with three games of the regular season left to play.
Just six points separate the Dragons from the last semi-final spot, and with fourth-placed Edinburgh travelling to Rodney Parade this Sunday, that Play-Off dream could be even closer by the end of the week.
"It's literally do or die for us," said Dragons skipper Tom Willis ahead of Sunday's clash.
"We've got serious ambitions of being in the top four, and if we want to make that happen, it's going to take a good win against a quality Edinburgh side who have been right up at the top end of the league for most of the season.
"There's still three games to go and we're taking it one week at a time but we're still keen to make that top four."
This year's Magners League campaign is proving to be the most open yet, with the introduction of the new Play-Off system keeping interest at a maximum all the way through the season.
Under last year's tournament rules, Willis and his Dragons would have little but pride to play for in the final three league matches. But this year's system means that Magners League glory could yet arrive at Rodney Parade.
"The intriguing thing from a supporters' point of view is that everything's still alive," added Willis, who committed his future to the Dragons by signing a new two-year contract earlier this week.
"There's lots to play for for the teams looking to get into the end-of-season semi-finals.
"I don't think the league's become any more competitive as it's always been very competitive, but I do think the Play-Offs have brought a fantastic new element to it. For teams like us at the Dragons, who are notoriously in the middle to the lower part of the table, we've now got a top-four opportunity.
"In the past, when you had to finish top of the table to win, we hadn't really pushed those buttons. But now we've got a sniff of getting into the top four and the semi finals and, if you get there, it's only two games to win the league.
"From our perspective, it's put a real focus and a real desire on doing well in the league. It's our bread and butter competition and it's one we want to do well in."
While much has been made of the Play-Off chances of both the Dragons and the Ospreys in the Welsh media, an equal amount of column inches has been given to the battle at the bottom of the table.
With just three of the four Welsh regions gaining Heineken Cup spots for next season, there has been plenty of talk as to who will miss out on top-flight European rugby.
At the beginning of the season, the Dragons would have been many people's favourites for that unwanted honour, but not now.
As things stand, the Scarlets look most likely to be playing Amlin Challenge Cup rugby next season, with the Blues the next side in immediate danger of losing out.
The Dragons are currently 10 points clear of the Scarlets and two ahead of the Blues and, despite it being mathematically possible for the Gwent region to miss out, Willis insists his side are focussing on matters at the top of the table.
"Down in here Wales, there's a lot of focus on that aspect and the fact that only three of the Welsh teams will make Heineken Cup qualification.
"We haven't thought too much on that. Of course I'd be lying if I said we didn't know it exits.
"But my attitude is eternally optimistic and it's about focussing on the semi-finals. If we can get close to that, then the situation with European qualification will take care of itself.
"So far, I think it's fair to say that we're happy with where we're at. Overachieved is a statement I've heard in the media but we've got a tight little group, we're starting to build a real confidence about us and we're certainly heading in the right direction."
If a win over Edinburgh on Sunday is preceded by anything other than a Scarlets triumph at home to the Blues on Saturday night, the Dragons' Heineken Cup qualification will be secured.
And having already beaten Rob Moffat's Scottish side at Murrayfield back in November and with a record of just one league defeat in seven fixtures at Rodney Parade this term, the omens look good for another precious victory for Willis and co.
"Beating Edinburgh away came at quite a critical point in our season. It was good to be able to go up there and grind out a win against a quality side away from home in difficult conditions.
"But there's been a lot of water that's gone under the bridge since then. We've had our moments since then - we've hit some speed bumps and also had a good run of games and results.
"There's plenty for both teams to play for. It's a week when Edinburgh are probably hurting after the challenging away game they had at Connacht recently.
"There's still three games to go and we're taking it one week at a time, but we've still definitely got ambitions of a top-four finish."
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