Dean Cornelius' 57th-minute header proved enough for the Sulphurites to progress into the third-round hat, but they were made to work hard for the win by their Northern Premier League visitors, the lowest-ranked side still left in the competition.
And Weaver admitted it had proved a testing night for his team, saying: "They played very well and threw the kitchen sink at us in the last 15 minutes. It could have been an easier night, that's for sure.
"They were full of resilience and have ability in their team and I hope they go on to get promotion this season.
"We wanted at to be a goal up half-time because we knew they would grow in belief the longer it stayed 0-0.
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