The Reds are now into the quarter-finals, while also alive in the Champions League and hunting a top-four finish in the Premier League.
Robertson, who scored in the 3-1 win, began: "We know we let ourselves down on Tuesday, obviously not the performance that we expect of ourselves.
"It's very rare you get to play the same team three days later, so it was up to us to go and put it right and we knew the FA Cup is do or die, you know you're either in or you're out. Thankfully, we're into the quarter-finals."
On his goal, the Scot also said: "Probably quite rare. It just opened up, and thankfully, I hit it clean. Most of them usually go over the bar during my time at Liverpool, so I'm glad I managed to get one in the net!
"Look, it set us on our way and then obviously to get a goal so quickly after it is pleasing.
"I thought we controlled the game quite well up until the 92nd minute when we lose a clean sheet, which is frustrating. But cup competitions are all about getting through, and we're delighted to get through now."
On turning the season around, he added: "Of course it can. We try to fight on all fronts. Obviously, the Carabao Cup didn't work out, the Premier League hasn't worked out, but we're still in two cup competitions.
"If we play like we did tonight, then nobody wants to play against us. But we've just been a little bit inconsistent this season, that's the fact of it.
"If we perform like we did tonight, then I think we'll be better off for it, and we can go far in both competitions."