Danny Wright headed the Robins into a 17th-minute lead with his 16th league goal of the season.
Dan Holman then got his 17th of the campaign six minutes into the second half as he calmly slotted home.
Welling, without a league win since October, rarely threatened as they remained in relegation trouble.
Speaking about an injury to Harry Pell, Cheltenham Town boss Gary Johnson told BBC Radio Gloucestershire:
Media playback is not supported on this device
"His ankle's come up a little bit and when your ankle comes up it looks like ligament damage, otherwise it doesn't blow up.
"But we've got a good medical team and we've just bought a new machine that helps you recover from ankle injuries - the minute we get an ankle machine that helps, we get an ankle problem!
"We don't know how long it'll be yet, let's hope it's not long-term, but he's going to be out for a couple of weeks at least."
Cheltenham stayed top of the National League as they extended their unbeaten league run to 19 games with a comfortable 2-0 win over lowly Welling.