Gareth Taylor rescued Wrexham from an FA Cup first-round replay at Lowestoft with an 89th-minute strike.
The visitors were resilient but Wrexham pressure finally told as Taylor got on the end of Wes Baynes' free kick.
The Ryman Division One North side - three divisions below Wrexham - belied their ranking for much of the contest.
Baynes had a free kick stopped in the 41st minute and Taylor's soft header 14 minutes from the end summed up the Blue Square Premier side's effort.
Wrexham manager Dean Saunders:
"The main thing is we're through to the next round. It was a difficult match.
"We lost last year to Eastwood who went on to beat a few teams and ended up getting promotion from their league and that's not easy to play against.
"They put everyone behind the ball and left the number nine up front on his own and they just hung on and hung on and hopefully they could nick a goal from the set piece from a corner or something like that or we did something daft.
"Chris Maxwell's had nothing to do all afternoon, frustrating watching, obviously because the chances went a-begging again.
"There was balls going across the goal and we never finished them off, but eventually I said to them with 20 minutes to go, 'They're not as fit as us, we're full-time training and they're not as fit as us and whether they can keep defending right to the 90th minute' and eventually we did grind them down and we got on top of them."
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