Rhinos begin their campaign with a sixth-round tie away to Widnes Vikings, but it is a tricky fixture on a ground where Leeds have suffered four successive defeats and were hammered 26-6 in February.
Leeds are also struggling for form, having won only three of their last eight games in Betfred Super League.
Briscoe, who scored five tries when Rhinos beat Hull KR in the 2015 Cup final, is confident they have the tools to get the job done, but admitted they need to be switched on from the kick-off.
“We don’t have a good record there,” said the winger, who also touched down in Leeds’s 2014 Wembley win over Castleford Tigers.
“In recent times we seem to have not performed there.
“We need to get our attitude and mentality right going into this game on the back of a few years of some pretty poor performances over there.”
Briscoe insisted Rhinos can’t use Widnes’s artificial pitch as an excuse.
He said: “I think it has been a combination of bad days, attitude and maybe the pitch has played a little part. But we have got a three-g training pitch and we get tackled on that so I don’t think it’s a major deal.
“For whatever reason we’ve just not performed or turned up and we have to do that.”
A Cup run could revive Rhinos’ season after a poor run which has seen them slip to sixth in Super League.
“We can put our league form to the side and concentrate on this,” said Briscoe.
“It is a new competition, it is knockout and we know we have got to turn up to make sure we are in the bag for the next round.
“The Cup is massive for us. At the start of any season you want to win silverware and the Cup is one of the three things on offer.
“Especially in the first round we’re involved in, we want to be progressing and hopefully we can get to Wembley which is a great experience for everyone involved.”
Briscoe added: “The start is key.”