The score line looked comfortable at the end, but the Mintcakes were stubborn opposition and it wasn't until the final ten minutes when Blues added two late goals that the game was safe.
Neil Baker made one change to the side that beat Mossley with Jessy Bavanganga starting in place of the injured Lewis Short.
Blues started on the offensive and Jordan Johnson had a free-kick deflected just wide in the 7th minute.
Town took the lead on 24 minutes when Rob Stevenson's cross was headed past the keeper from close range by Tim Grice as he scored for the fourth consecutive league game.
Soon after Stevenson and Charlie Walford both had shots saved before Leek were awarded a penalty for a foul on Jordan Johnson in the 36th minute.
Grice took the spot-kick but held his head in hands as he placed his effort wide of the post.
The visitors improved a little after the break but Blues young defensive line held firm restricting them to a few speculative efforts that were unable to trouble Danny Roberts who barely had a save to make in the game.
Stevenson had a good effort on 63 minutes which curled just wide off the post but it wasn't until the 81st minute that Blues finally killed off the game when Billy Sass-Davies scored with a towering header that went in off the underside of the bar.
Substitute Aaron Bott went close a minute from time when he diverted Walford's cross over the crossbar, but he did score in added time when Stevenson's parried shot reached him at the far post and he stooped to head home.
View the match photos here.
Line-up:
1. Danny Roberts
2. Niall Green
3. Jessy Bavanganga
4. Charlie Walford
5. Billy Sass-Davies
6. Jack Sanders
7. Niall Maguire
8. Billy Reeves
9. Tim Grice (14. Aaron Bott, 83)
10. Jordan Johnson
11. Rob Stevenson
Unused subs: 12. Sam Hall, 15. Dan Trickett-Smith, 16. Lee Wilshaw 17. Ben Chapman
Attendance: 313
# | Team | Pl | Pts |
1. | Atherton Collieries | 38 | 82 |
2. | Radcliffe FC | 38 | 75 |
3. | Leek Town | 38 | 74 |
4. | Colne | 38 | 73 |
5. | Ramsbottom United | 38 | 72 |
6. | Runcorn Linnets | 38 | 72 |
7. | Prescot Cables | 38 | 69 |