Conceding a late goal to Young Boys in a 2-2 draw at Anfield last night means Brendan Rodgers' side have to win against already-eliminated Udinese in a fortnight to guarantee progress to the knockout stage.
The Reds could also get through if they avoid defeat and at least match Young Boys' result against Group A qualifiers Anzhi but a loss puts them out even if the Swiss club are beaten.
"We're still confident that we can qualify. We need to go to Udinese and try to win the game," said Cole.
"It is very complicated but if we win the game that puts it all to bed.
"It will be tough over there but I'd imagine that we will go there with a strong squad.
"Liverpool Football Club need to be playing European football after Christmas."
Cole had one of his best, if not the best, games since his 2010 free transfer from Chelsea and scored his first Reds goal for 19 months to put Liverpool 2-1 up against Young Boys, who denied the former Chelsea player his spot in the limelight with an 88th-minute equaliser.
He has struggled to find any form since then and after a season-long loan spell with Lille a failure to make an impression on the first team placed his immediate future in doubt.
But having recovered from injuries earlier in the campaign the 31-year-old still feels he can succeed at Anfield, although Rodgers will have the final say on that.
"It was nice to be out and playing. It's been a long time since I got one in front of the Kop" said Cole, who is confident a hamstring niggle which forced his substitution will not stunt his progress.
"It was my third game back from injury and I've felt stronger in every game I've played.
"But there's no rocket science about it - if you're a footballer and you return from injury for one game and after that you play a second, you feel a little bit better and in the third game, you feel better still.
"You have more fitness and match strength. I'll continue to try my best and we'll see what happens.
"I just need to keep going and keep plugging away."
Despite failing to so far convince Rodgers of his value to the side Cole is fully supportive of the direction the Northern Irishman is taking the club.
"The manager is doing a great job, the young players have been brilliant and the lads who have come in and have done really well and we just need to keep moving on and going forward," he added.
"We need to learn from these experiences so that if we're in that situation again that collectively we do a lot better
"There were a lot of positives to take from the game but overall it's a feeling that we could have done better."
Fellow midfielder Jonjo Shelvey, who played as a nominal striker and grabbed his fourth European goal of the campaign, gave his backing to Cole after receiving the cross from him for the opener.
"Joe has had a tough time since he's been at the club - he got sent off in his first game a while back - but he did really well," said the 20-year-old.
"He deserved his goal. He works hard day in, day out, and is a credit to himself."
Shelvey admitted he was slightly taken aback to discover he was to be pushed further forward against the Swiss.
"The manager pulled me on Wednesday to tell me and it was a bit of a surprise," he added.
"It was unfamiliar for me but it was good. I'd play at left-back if it meant getting a game for Liverpool.
"It is just a pleasure to be coming in and working for Liverpool every day.
"Obviously I will always look to go out on the pitch and score goals - sometimes they fall and sometimes they don't and recently in the league I've had a few more chances and not put them away.
"But if you keep getting in there the goals will come."