The Hammers face an uphill task to progress after throwing away a 2-0 lead in the first leg to draw 2-2 following James Collins' sending off.
Hammers are underdogs
As the home side, and with two away goals in the bag, Astra will start as favourites particularly given they saw off the likes of French giants Lyon and Croatian big guns Dinamo Zagreb in the group stages of the competition last year.
Jenkinson is one of few first team players in a skeleton squad sent to Romania by Bilic as he cannot play in the Premier League opener against parent club Arsenal on Sunday.
And he is bullish about his side's chances, telling the assembled media, as reported by West Ham's official website, he is confident they will win the game.
'Europe is a big thing for everyone'
The 23-year-old insists he wants to progress to give the club's fans European football in their final season at their Boleyn Ground home of 112 years before they move to the 54,000 seater Olympic Stadium next summer.
"He's (Bilic) had to leave a few players at home with one eye on Sunday's game but the players that have come are obviously top players otherwise you don't play for West Ham in the Premier League," Jenkinson said, as reported by whufc.com.
"We've got a good squad and we'll have a good team out and we've got more than enough ability to go and win the game.
"Of course you want to progress, we'd love to go through and of course with regards to the fans they'd love us to be in Europe for the last season at the stadium, it's a big thing for everyone."
Europa League: the pros and cons
It has been a while since West Ham were last in Europe. They lost a UEFA Cup qualifier 4-0 to Palermo nine years ago so that tells you it is not something that comes around very often in east London.
Therefore it is understandable why many fans are hopeful of seeing the club progress to the group stages, so they can aptly play some big European games under the lights at the Boleyn for the last time.
However, there is the obvious hindrance that comes with being in the competition.
If West Ham were to go all the way in the competition it would have added a staggering 23 matches to their season.
Add on top of that the travelling and relentless Thursday, Sunday schedule and it can take its toll on even some of the strongest squads such as Everton and Tottenham Hotspur's last season.
Past history tells us that teams competing in the new format of Europe's second tier competition often suffer domestically back in the Premier League.
With everything that is at stake for West Ham and new boss Slaven Bilic this season it could prove to be a competition that presents too much risk for what is gained.
But there are still a number of dreamers out there, and after all isn't dreaming what football is all about?
They will point to the fact that there is always that golden carrot of winning the Europa League trophy and qualifying for next season's Champions League, something that is well beyond West Ham's current reach through the league.