Substitute Jonny Hession’s strike 20 minutes from time ended stubborn Eccleshall’s resistance and took Nelson seven points clear at the top of the NWC First Division.
Both teams made a tentative start, with the Blues playing with a stiff breeze behind them.
Gradually the Admirals stepped up the pace, but chances were few and far between.
Ed O’Neill was prominent down the flank, but his promptings failed to trouble the visiting keeper Christian Middleton-Hill.
A dangerous inswinging cross from Lydon Hodgkiss was tipped over the bar, while two half chances fell to Peter Wright in quick succession but were squandered.
Danny Mahoney went on a long run through the defence, but his final pass to Wright was cut out.
The Blues continued to press the overworked Eccleshall defence, but couldn’t make the breakthrough, as an inswinging cross-shot from Mahoney deceived Middleton-Hill but bounced off the top of the bar to safety.
At the other end, a rare chance fell to James Morrow, but he sliced wide of Ben Woodhead’s goal.
At the start of the second half, Sakeel Abdullah came o for Wright, who had pulled up injured just before the break.
This added extra pace to Nelson’s attack, and Abdullah had an immediate effect, forcing a corner.
In was one-way traffic, but the Eccleshall defence remained stubborn.
Matt O’Neill found the side-netting from a half chance, before Hession replaced the tiring Michael Morrison.
Within minutes of entering the fray, Hession was picked out by Robert Grimes, and he cooly beat the keeper from close range.
Mahoney, who had been pulling the strings in midfield, had a goalbound effort blocked as Nelson looked for a second goal, but one proved enough to claim victory, as the Blues recorded a third-successive clean sheet in the league.
That leaves Nelson seven points clear of second place Formby, with nine games remaining, although Formby have three games in hand.
They are also nine clear of third-place Atherton Collieries, who have played two games fewer, 13 clear of 1874 Northwich, who have four games in hand, and 16 clear of Hanley, who have five games to make up on the Blues.
But points in the bag are crucial at this stage of the season, and Nelson go in search of more tomorrow when they host Widnes Vikings, kick-off 3 p.m.