The Bucks scored twice in second half stoppage time to beat FC United of Manchester 3-1 on Saturday.
They are set to find out who they will face next in the competition – maybe a League One or Two side – in tonight’s televised draw, which starts at 7pm.
“The lads showed great character to get the win,” said Edwards.
“I am really excited – this is where we wanted to get to, to win the three ties and get into the proper rounds.”
Before the 3-1 victory against FC United – Marcus Dinanga (two) and Jordan Lussey getting the goals – Telford beat Evo-Stik Northern Premier teams Barwell and Stafford Rangers.
They required a replay to get past Rangers and Edwards said: “It’s been a bit of a rollercoaster ride already.
“In that first game against Stafford, we were down to 10 men with 10 minutes to go and that could have been it, but we managed to get the draw and then win the replay.
“The lads have shown a lot of character and we’re there, we’re in the hat and we’ll all be raring to see who we’ll be facing in the first round.”
Dinanga opened the scoring in the 52nd minute, in front of 1,451 fans at the New Bucks Head.
United’s Craig Lindfield equalised before Dinanga was fouled for a penalty in the 92nd, which he put away. Lussey put the icing on the cake with a smart finish in the 94th minute.
“It was nice to win in the manner that we did,” said Edwards.
“The game was pretty tight to be honest, Jas Singh made some good saves to keep us in it in the first half.
“We went ahead but then FC United cause us some problems after making some changes, they really went for it.
“They got the equaliser so we made some changes as well to try to get the goals and we managed to do it.”
Dinanga’s brace took him to 10 goals for the campaign. The tally means he has already surpassed the Bucks’ top goalscorer from last season, Josh Wilson – who got nine.
The Burton Albion loanee has set himself a target of at least 30 by the end of the term.
Edwards added: “Marcus has done really well.
“He brings that goal threat but it’s not just him, we’ve got a fair few players who are capable of getting the goals for us.
“When you have a threat like that, you know you always have a chance if you can keep it tight at the back.”