Huddersfield, third in League One, always had plenty in reserve against the Conference South pacesetters despite a slender 1-0 lead at the break.
Booth opened the scoring and grabbed the all-important third to put Huddersfield into the second round for the first time in three years.
The Terriers made a sluggish start but once Mark Hudson, Andy Holdsworth and Chris Brandon started driving them on it was one-way traffic for the League One promotion hopefuls.
Danny Adams lashed wide before Gary Taylor-Fletcher, back in the side at the expense of top scorer Pawel Abbott, cut inside and forced Jamie Turner into a diving save.
Welling showed little in attack and most of their threats were self-inflicted by the home defence.
The breakthrough came in the 27th minute when Danny Schofield's corner was powerfully headed home by Booth.
Huddersfield pressed forward and Adams' curling free-kick deflected off the wall for a corner, before Hudson ended up on his back when he tried an ambitious overhead kick in front of goal.
A long clearance by keeper Paul Rachubka allowed Taylor-Fletcher to test Turner again and in first-half stoppage-time Booth headed Hudson's swirling cross just wide and Brandon's drive was again blocked by Turner.
Huddersfield sealed victory in the 50th minute with a Schofield penalty. Brandon was tripped by Matt Lee and Schofield slotted his seventh goal of the campaign.
The unlucky Taylor-Fletcher was denied again by Turner before Abbott made his entrance after Brandon limped off.
Booth's second came in the 70th minute when the Welling defence allowed Rachubka's clearance to bounce. Booth remained patient, turned and smashed the ball past Turner.
Holdsworth made it four when he fired in from Schofield's cross after Turner's one-handed save from Abbott.
Welling grabbed a consolation through skipper Chris Moore's header seven minutes from time and were not disgraced.