Premier League Brentford travel to Blundell Park on Tuesday for a fourth-round tie hoping to avoid becoming the Mariners' latest victims.
Grimsby, sixth in English football's fourth division, produced one of the great cup shocks by beating Manchester United in the second round after a marathon penalty shoot-out and followed that up by knocking out Championship crisis club Sheffield Wednesday.
But Brentford head into the last 16 buoyed by their 3-2 win at home to Premier League champions Liverpool on Saturday, with Andrews saying: "The bench has made a huge impact in recent weeks - goals from it, courage, personality and experience - so they need to be rewarded and the squad will be utilised."
Grimsby's push for promotion has faltered recently, with last weekend's 3-2 loss at Crewe their second defeat in three league matches, but Andrews added: "We're really up to speed on their key players and the system they play.
"To get to this point in the competition is significant for them. They've achieved a lot this season and are a very well-coached team. We have to be very respectful of them, but we'll be ready."
Wrexham welcome Cardiff in Tuesday's other tie in an all-Welsh clash that will see the two clubs meet for the first time in 21 years.
The Dragons have enjoyed three successive promotions under Hollywood owners Ryan Reynolds and Rob McElhenney, with Wrexham aiming to reach quarter-finals for the first time since the 1977/78 season.
"It's a Welsh derby and the chance to get into the quarter-finals for only the third time in our history so it's an exciting evening ahead," Wrexham manager Phil Parkinson told the BBC.
"I had a chat with the lads about our history in this competition and the rivalry between Wrexham and Cardiff.
"I don't think it needed too much explaining because everywhere I've been going in the last few weeks, supporters have been speaking to me about that and I'm sure the players have been the same."
Cardiff, top of League One, knocked Premier League Burnley out in the previous round.