The 18-year-old, who scored within 10 minutes of making his bow for the club in the League Cup last season, put in a composed performance in his first European outing which saw the Reds draw 1-1 in Bordeaux.
For over an hour he played alongside fellow academy graduate Pedro Chirivella, who made an earlier-than-expected first-team bow after injury to Kolo Toure, and by the second half the pair looked as comfortable as they did playing in the under-21s together.
Local-born Rossiter's rise through the ranks has accelerated over the last year and he is keen to seize the opportunity offered by Brendan Rodgers, who is likely to use him again for next week's Capital One Cup visit of Carlisle and also in forthcoming European fixtures.
"It was a tough test and I really enjoyed my European debut," he said.
"I loved it out there. I've been working most of my time at the academy to try to push into the first team and I got my chance. Hopefully it comes again.
"Young players want to do well and we want to go that extra mile.
"I've played with Pedro for two or three years now so I know how he plays and he knows how I play and I thought he did very well – the same with Cam (Cameron Brannagan, another academy product given a late debut) when he came on.
"To be honest I was a little bit nervous when I found out I was starting a few days ago but you just have to go out there and enjoy it and try your best – that is all you can do.
"I had to come off with a bit of cramp, which I'm disappointed about, but the manager said, 'great experience' and I can only take that forward and hopefully get better.
"I was just disappointed we couldn't get the win but I think a point away from home is good.
"We have three home games left and if we get three wins from them it will be good for us."
Although still a teenager, Rossiter has been around long enough to know he is unlikely to keep the likes of James Milner and Lucas Leiva – among six first-teamers left at home – out of the side for Sunday's visit of Norwich but that has not dimmed his ambition.
"I'm not going to get carried away with the performance, I know there are a lot of players to come back and I am still only young," he added.
"I've just need to keep improving and if I get my chance again take it."
The only downside was his family could not join him at the Nouveau Stade de Bordeux and had to watch his European debut on television.
"With work commitments they couldn't get over but they are all made up for me," he said.
"When my dad and family used to take me to football throughout the years it is all for moments like this."