Stevenage fought back though to lead at the break with goals from Tim Sills, his first since joining from Torquay, and Chris Beardsley.
Second half strikes from Michael Bostwick and Yemi Odubade made it safe despite Welford twice going close.
Stevenage manager Graham Westley was far from happy with his sides' performance when he told BBC Three Counties Radio: "I was disappointed in us to be honest.
"I think that the level of work-rate that we showed this afternoon was below the level that we are capable of and I don't think our dominance of the game was consistent enough.
"They (Dover) came and put in a spirited effort, but I felt we are better than we were this afternoon.
"It was disappointing to concede and we didn't do enough damage going forward."
Source: BBC Sport
Source: BBC Sport