City have struggled in the first two months of the season but Jones hasn't been unhappy with performances.
They didn't make easy work of their FA Cup tie at home to Leiston of the Bostik Premier and needed two very late goals to secure a 4-2 win to confirm their passage through.
Jones paid tribute to his squad for playing until the end - and singled defender Joe Oastler out for special praise.
"It was a fantastic win, I'm really pleased for the players," said the manager in a video interview on the club's website. "We have had a lot of disappointment in the first few weeks of the season.
"We have been playing OK but not getting the results but today we did and I am delighted for them and really pleased for the club to be in the hat for the next round, even though we left it very late.
"We started the game quite poorly, we were quite slow. It was a topsy turvy match.
"Joe Oastler was superbly for us today, real good. He scored from a set play and it's nice for us to be on the other end of one - we usually conceded from them!
"We came on strong in the second half and we showed a lot of character.
"I was pleased for the fans too and they had some important late goals to cheer. It's great to go through."