The Turkish club arrived at Anfield for Wednesday's last-16 second-leg tie with a 1-0 lead but were swept aside 4-0 by Liverpool.
Six-time European champions Liverpool can now look forward to a quarter-final with holders Paris Saint-Germain, who knocked them out last season.
It was a rousing result for Liverpool, who have struggled to defend their Premier League title and are currently fifth in English football's top flight - 21 points behind leaders Arsenal.
"There was a lot at stake, that was pretty obvious, so we had to show how much we wanted it," said Van Dijk.
"I think occasionally we have been playing good games but not consistently on a regular basis and it's been frustrating for all of us and for the fans as well."
The 34-year-old Netherlands defender added: "It is very hard, but that's the hardest thing in life, doing it each and every three or four days. If you do that there's a big chance you become successful.
"That's what you strive for, that's what we had last year, and that's what we try to reach, but you need multiple factors to go your way and now we don't get carried away.
"We have been having good performances this season and we haven't been able to build on it the next time.
"This was a demanding game physically and mentally and it is down to us to do all the right things to be ready for an early kick-off at Brighton (on Saturday)."
Victory over Galatasaray was badly needed by Liverpool following a lacklustre draw with relegation-threatened Tottenham at Anfield on Sunday, which led to rare boos from frustrated Reds supporters.
"The fans can probably say if we bring that every week on the pitch, then there is no issue and we can say if the fans bring that every three or four days then there is no issue either," said Van Dijk. "It is about doing it together.
"This season as players we take full responsibility for a lot of performances when we have not been good enough.
"But for me, there is no point to speak about it right now after we qualified for the last eight because that is still a very good thing."