The Pilgrims head to National League North rivals Brackley Town this afternoon (KO 3pm) with a place in the third round of the competition at stake.
"It's a competition at our level where we have a genuine chance of getting to Wembley," Elliott said.
"We've seen plenty of teams in our league get to the final and win it.
"There's history with Boston going there as well, so it'll be nice to do it."
The Pilgrims' 2-1 defeat to Wealdstone in the 1985 final, with Chris Cook on target for Boston, is the closest United have come to winning the competition while today's hosts - then with midfielder Shane Byrne in their ranks - won the competition in 2018.
The Trophy has been something of an afterthought for Elliott during his time with Boston.
In 2017-18 they exited the competition following a 2-0 replay defeat at Kidderminster Harriers, with focus very much on pulling the side away from relegation at the time.
A 5-0 thrashing of Ashton United was followed by a 3-0 thumping at Wrexham the following year while a second-string side was sent to Atherton Collieries in 2019-20 as players were rested for the following week's FA Cup second round tie at Rochdale.
Last season saw Boston beat AFC Fylde on penalties before a shootout defeat to Chesterfield at the Jakemans Stadium brought a halt to the season.
But following the Pilgrims' FA Cup exit at Stratford Town, Elliott would love to put a cup run together.
"We didn't manage to hit our target in the FA Cup, we were a game short," he said.
"So we have another chance in the FA Trophy to create something.
"I could have picked a lot of other teams to play against, but it is what it is and to get there (Wembley) you've got to beat good teams. That starts Saturday."
Elliott joked 38-year-old Paul Green was 'on his last legs' after Tuesday night's win over Darlington after playing his third match in 10 days. However, the Boston boss doesn't want to make unnecessary changes to a side unbeaten in their last two matches.
"We've got a few bumps and bruises but we're not changing for the sake of it," he continued, suggesting he may be sticking with the five-at-the-back formation that has earned a point against Chorley and three against the Quakers.
"It suits a lot of players.
"Joe (Leesley) and Fraser (Preston) lose out a little bit but it does show we can be flexible who we play against, and you need that in this league.
"We've played well in a different shape which has been pleasing, the players getting some more confidence from another good performance.
"Let's hope we can get another good result away at Brackley."
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