Saints opened the scoring through Stevie May after just four minutes but the third-round tie did not prove to be quite as simple as their impressive start may have suggested.
The Perth side had to wait until four minutes to go before substitute Gwion Edwards put the tie beyond Accies, with May completing his double deep into injury time.
Wright said: "I thought we were quite comfortable. We worked really hard and we limited them to few opportunities.
"I think 3-0 is probably a fair reflection in the end of the dominance we had in the game.
"I'm pleased we got the second goal and then the third goal was a fair reflection of the game."
Saints headed into the tie without a win in their past four matches, while Scottish Championship leaders Accies had won seven of their past eight games.
Wright added: "Any time in the cup when you're drawn away from home and against a team who are doing well, it's difficult.
"There are no easy ties in the cup. People were probably looking at the fact that we hadn't won in four games and thinking does it put added pressure on us?
"But even coming here having won your last three or four games, it's still going to be a difficult tie.
"They are a good side, with a lot of good players. But we were very professional in how we went about it and deserved the win."
Accies boss Alex Neil said: "I thought St Johnstone deserved to win the match but I don't think it was a 3-0 scoreline to be honest.
"I've got absolutely no complaints about my players - I thought they gave it their all.
"We just weren't good enough to win the match."