Tuesday's 3-0 win against Accies brought to an end a run of four games without a victory for Saints.
Wright now believes the Perth side can take confidence from the win into this weekend's league clash against Partick Thistle.
He said: "Getting the win will help to kick-start our league campaign again.
"But it will be a hard game. Partick Thistle are always a hard team to play against."
Thistle head into the weekend action following a 4-1 cup loss to Dundee United and Wright is wary of a backlash.
He added: "It's always hard when you get beat midweek in the cup.
"You're always looking to bounce back in the next game so we will be wary of them."
Saints were drawn against Celtic's conquerors, Morton, in the last eight of the League Cup.
But, despite both Old Firm clubs being out of the running, Wright was reluctant to look too far ahead in the competition.
He said: "We will just be happy to get into the next round.
"You can't say you're going to win it or you're going to get to the final because these games are always hard."