The hosts had the chance to move up three places and, although they failed to do so, they have plenty more games in hand on everyone above them.
Ritchie Jones brought done Nathan Stone to concede the 14th-minute penalty with which Gareth Heath put Leiston ahead.
Thurrock had a fair share of the game but conceded a soft goal 90 seconds after the interval, a simple near-post header by David Head from a corner.
Emmanuel Osei broke from halfway to make it three in the 57th minute, following up after David Hughes turned his initial shot on to a post.
It was a harsh scoreline for the home side, who kept battling but were twice foiled by the woodwork in their search for a consolation goal. Matt Turpin and Danny Green hit the same post in the space of a couple of minutes.
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