The Reds had made their worst start to a league season in 100 years at the five-game stage, but their form has steadily improved since then and they are now unbeaten in seven league games.
Liverpool still sit eight points behind fourth-place West Bromwich Albion, but Carragher believes that teams will drop points all season, leaving opportunites for them to reduce the deficit.
"We look as if we have not made a good start, but with winning on Saturday, we are not too far behind the top four," the 34-year-old told the BBC.
"It is still up for grabs. Beside the top two, and even Manchester United lost on Saturday, a lot of teams are dropping points all over the place.
"From our own point of view, we now want to kick on and make a challenge for that top-four place because it seems that everyone is throwing it away."
Liverpool, who now sit 11th in the table, will face Swansea City in their next league game.