The Robins went into the lead in the opening minute thanks to a Billy Waters finish, with Rob Dickie doubling his side's advantage eight minutes later.
Things then went from bad to worse for the visitors just three minutes into the second half, with Adam Lockwood receiving a red card for pulling back the last man before Waters took advantage with his second of the game. Jack Munns and James Rowe finished the scoring in the final 10 minutes.
Forest Green dropped to second after a 1-1 away draw to 18th-placed Welling United, who went ahead through Michael Bakare before Kurtis Guthrie equalised for the visitors.
Boreham Wood secured an important 2-1 away win against Bromley with a Luke Howell brace. Moses Ademola threatened a comeback for the home side with a goal shortly before half time, but the guests held on for victory.
Elsewhere, Woking and Braintree Town fought out a 1-1 draw at the Kingfield Stadium, with Sean Marks opening the scoring for the visitors before John Goddard equalised with 12 minutes remaining.
Results in full: Barrow 1-3 Grimsby Town (1-1), Bromley 1-2 Boreham Wood (1-2), Cheltenham Town 5-0 Guiseley (2-0), Chester 3-1 Kidderminster Harriers (1-1), Dover Athletic 1-2 Eastleigh (0-0), Macclesfield Town 3-0 Altrincham (2-0), Welling United 1-1 Forest Green Rovers (0-0), Woking 1-1 Braintree Town (0-1).