The European champions have won just one Premier League game this month, and were also knocked out of the Capital One Cup by Swansea City last week.
Only an 83rd-minute equaliser from Fernando Torres stopped Chelsea being eliminated by the League One outfit, and Cahill says Benitez inspired the team to avoid defeat.
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"There is a lot of pressure from the fans but we needed to just try to do the job," he told reporters.
"Everyone was aware how important the game was. Ideally, we would have gone in at half-time a couple of goals up but that wasn't the case.
"There were a few words said in the dressing room from Rafa and then between the players themselves. Everyone knew we were below par. We spoke about that and we needed to do something about it in the second half.
"In the end, we grafted it out on a horrible day and we now need to show our quality at home. It was the character we needed to dig in and just get the result we needed.
"Obviously, we wanted the win but at least we have taken it back to the Bridge and hopefully we can finish the job there."
With World Club Cup commitments already causing a backlog during their January fixture schedule, a fourth-round replay now sees Chelsea play a minimum of 58 games this season, and Cahill admits an extra game is something he and his team-mates could do without.
"Having a replay is not ideal," he continued. "There are so many games, it feels like you're only just recovering and then there is another game again.
"You're trying to get your full recovery in before the next game and it's hard when there are not many days to do that. Sometimes, you do feel leggy."