The Shaymen eased to victory at fellow Skrill Premier side Nuneaton thanks to a goal in each half from Lee Gregory and Scott McManus.
Nuneaton rarely troubled the visitors, who produced a solid defensive display to reach the first round for the second time in the last six years.
Aspin said: “It is always going to be a tricky tie if you get a team in your own league away from home but I thought my players gave a really good professional performance, defended well as a group and obviously scored two great goals at the other end.
“It is a really satisfying performance. I am happy with what the players gave me in terms of effort and commitment. They did what they needed to do to win the game.”
Gregory’s ninth goal of the season gave Halifax the lead after 17 minutes when he acrobatically swept Adam Smith’s cross past James Belshaw in off the far post.
The Town striker was then called into action at the other end four minutes later to clear Louis Moult’s shot off the line from a Nuneaton corner.
The hosts only had a couple of long-range efforts to show for their attempted response until the interval, though, with Jon Adams and Adam Walker both going close.
McManus showed them how it should be done just after the hour-mark to effectively end the tie as a contest when he ran with the ball from the halfway line before unleashing a left-foot shot from 25 yards that beat Belshaw with the help of a deflection.
McManus nearly added a third late on but his low shot at the far post from Smith’s cross was saved.
Nuneaton Borough: Belshaw, Richens (Pearson 88), A Bell, Streete, Dean, Adams (M Bell 72), Walker (Hibbert 68), Armson, Moult, York, Trainer. Unused substitutes: Sleath, Taylor, Gordon, McNamee.
FC Halifax Town: Glennon, Toulson, Roberts, Ainge, Lowe, Smith, Maynard, Pearson, McManus, Gardner (Worthington 72), Gregory. Unused substitutes: Senior, Wilson, McReady, Marshall, Holsgrove, Johnson.
Referee: C Kavanagh.
Man of the match: Marc Roberts.
North Ferriby gave Alfreton Town a shock by seizing a 15th-minute lead but the visitors’ superior finishing told as they won 3-1.
The opening goal arrived when Jamie Yates’s corner was played back to him and Nathan Jarman headed home the cross. Jake Speight equalised in the 39th minute and John Akinde put Alfreton ahead on 69 minutes with a low drive. John McGrath curled in his side’s third, deep into stoppage time.
Bradford Park Avenue and Kidderminster Harriers will replay tomorrow after a 1-1 draw in wet and blustery conditions at Horsfall Stadium yesterday.
Alex Davidson’s curling shot gave Avenue a 26th-minute lead against opponents one league above them, but Jack Byrne equalised nine minutes later.