The Wetherby Road outfit end the opening weekend of the new season top of the table having beaten Southend United 4-0 away from home in what was their first-ever outing in the Football League.
A brace from Jack Muldoon sandwiched strikes by Lloyd Kerry and Aaron Martin as the 2019/20 National League play-off winners picked up where they left off last term.
“It’s amazing, an amazing feeling,” said Weaver after watching a side comprised entirely of players who got Town promoted fire the club into pole position.
“We’re not getting carried away because momentum is when you really string results together. It might not always be 4-0 wins and I don’t want expectations to go through the roof irrationally, but we can be pleased that it’s a great start.
“I feel immense pride. That starting XI were all with us last season and I’ve said all along that those players deserved and fully justified the chance to have a go at the Football League.
“The early indication is that we’ve stood by some good characters.”
Town headed into Saturday’s contest full of confidence following last month’s Wembley triumph and a penalty shoot-out victory over Tranmere Rovers just seven days earlier.
By contrast, Southend are very much on a downward trajectory and beset by off-field problems, yet Weaver insisted there was never any danger of his players taking anything for granted at Roots Hall.
“Momentum is with us at the moment, but you don’t get ahead of yourselves in this game,” he added.
“I’ve played in and managed enough games to realise that you have to be prepared to start all games with the same ambition and the same motivation and not be blasé about anything - and the lads’ attitude was outstanding.
“Everything that we’ve worked on over a long period of time was for occasions just like this.
“Everyone knew their roles and just carried them out brilliantly - and we had an end product, which was so pleasing.
“I’d have been happy with a scruffy 1-0, but this is a fantastic result.
“It couldn’t have gone any better. It’s a terrific start.”