The visitors looked capable of causing an upset after Dan Fitchett's equaliser midway through the second half, but three late goals gave Vale a flattering margin of victory.
Adams said: "In cup competitions it doesn't matter if the final score is 1-0 or 4-1 because all that matters is we are in the next round.
"I'm happy with the result but not happy with the performance. I thought both sides were flat in the first half and fortunately we went in a goal up.
"We wanted to up the tempo in the second half, but we didn't do that and Salisbury had a really good go at us.
"At 1-1 it was understandable our fans would get nervy because when you look at Port Vale's history you will see it is littered with FA Cup defeats to non-league sides.
"But we've managed to produce some moments of quality in the final third and go on to win the game.
"The prestige of being in the third round is always good for lower league clubs and hopefully we will get a good draw and bring in some money, which always helps.
"I know our fans would like to play Stoke City, but an away game at either Manchester United, Arsenal or Chelsea would do me fine."