Both teams came into the only match in the division on the back of good wins in the Buildbase FA Trophy last Saturday, but the visitors needed points to move away from the bottom end of the table as they started the evening just four points clear of bottom side Hitchin Town.
The Suffolk hosts had a great chance to take an early 9th-minute lead when Joe Marsden was brought down in the area. However, Will Davies` penalty came back off a post and bounced to safety.
The first half was entertaining with chances at both ends but finished goalless.
The Marketmen had the first chance in the second half 5 minutes in but Kyle Hammond sliced his effort wide.
At the other end, the visitors` experienced keeper Marcus Garnham saved well from a Siju Odelusi free-kick and a Brendan Ocran shot but was finally beaten in the 82nd minute when Leiston were awarded a second penalty.
Will Davies was brought down by Garnham inside the box, and it was Finlay Barnes this time who took responsibility and he duly converted to make it 1-0.
The hosts held firm for the remaining minutes to move into the top five.
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