The Arrows had only made one appearance in the first round of the competition before this season, a 1-0 defeat at Stevenage in 2002.
They were given a bye through to the first qualifying round and then kicked off their campaign with a 3-1 win over Chatham Town in September before going on to beat Staines, Hitchin and Blackfield & Langley, ending their 10-year exile from the first round proper.
A narrow 2-1 victory over Bishop's Stortford came before a televised shoot-out victory against Harrogate in the second round and Ray says his team are ready for the biggest game in their history.
"There is a lot of pressure of course," he said.
"This is a big deal for us. It's our first-ever appearance in the third round so we've had a lot of people wanting to talk to us. It's been nice, but it's also been pretty hectic.
"But the players have been fantastic. I've had no trouble with anyone, they've all stayed pretty grounded. At the end of the day this kind of thing doesn't happen all the time."
The Ryman Premier Division club, who are the lowest-ranked side left in the competition, are enjoying a rich run of form.
Despite drawing their last six matches in all competitions, they are unbeaten in 12 and have lost just one in their last 18 outings.
Boro are enjoying a fruitful season themselves, currently positioned joint-third in the Championship, they are just two points adrift of the automatic promotion places.
Ray says although Tony Mowbray's side pose a different threat to what they are used to, the minnows are buoyant heading to the Riverside.
"We know it's a different level up in the Championship. Middlesbrough are a tough side to beat, for any side. They've been in good form themselves this season," he added.
"But we have to be pleased with our current run and what we've managed to do so far this season. At our level it's a very good record.
"In our division we've been very difficult to beat. Whether it's Middlesbrough or whoever we have to hold on to that mentality."
Hastings have barely named the same starting XI throughout their impressive run and Ray believes it has made his side more competitive.
"Everyone is fresh. We've changed things more or less every game. At the same time though it's given everyone a chance to impress ahead of the weekend," he added.
"They've all played their part in this competition as well as the league so it's been nice to have a fight going on for places.
"You've got to be consistent as well though and I think recent results have shown that we have been able to get it right.
"I've got a decent set of lads and there are some good characters in there. They can all do a job for us."
Ray could have a dilemma ahead of kick-off having confirmed he has a near fully-fit squad to choose from.
His only concern is over goalkeeper Matt Armstrong-Ford, who has been struggling with a persistent knee problem, and Ray is keen to bring in a replacement before the game.
"We've actually been speaking to Newcastle this week about getting Jak Alnwick in on loan in time for the game. Nothing is confirmed yet but we're hopeful of sorting it before we travel."