Hughes was appointed Southampton boss on Wednesday following the sacking of Argentine Mauricio Pellegrino and his side, 18th in the Premier League, struggled in the opening 45 minutes against League One promotion chasers Wigan.
Wigan, who had beaten three Premier League teams in the Cup including Manchester City in the previous round, could not convert their chances, however.
The home side went close in the 15th minute when Nathan Byrne’s deflected shot drifted goalwards but Cheyenne Dunkley was unable to connect from close range.
Gary Roberts then hooked over the bar from a promising position and Southampton keeper Alex McCarthy had to punch away an inswinging Max Power corner that could have sneaked in the near post.
Yet it was a different story after the break as Southampton came out strongly and their superior quality began to show.
Hojbjerg forced a fine save out of Walton with a powerful header from a corner and then from the resulting corner came the 62nd minute opener.
Hojbjerg made a positive run from deep to convert Dusan Tadic’s ball with a side-foot finish for his first goal for the club.
Southampton had a chance to wrap up the tie when they were given a penalty with 18 minutes remaining but Manolo Gabbiadini’s spot kick, awarded after he had been brought down by Dan Burn, was brilliantly saved by Wigan keeper Christian Walton, diving to his left.
Southampton’s second goal came in injury time when Cedric finished off a swift counter-attack, led by Nathan Redmond, with a calm finish into the far corner.
“It was an important day for the players today,” said Hughes.
“I have been really impressed with the quality of the talent I have. They have lost a bit of confidence recently but they are good players. We stood up to Wigan. It was a statement of intent and a start. We have work to do in the Premier League, but we will enjoy this moment.”
For Wigan, who had also beaten West Ham United and Bournemouth on their Cup run, the task is now to secure promotion back to the second-tier Championship but their manager Paul Cook was proud of their effort.
“You can’t ask any more from your players. The first half we were excellent. We expected that response from Southampton and we wish them well in the semi-finals. We were playing some great players and in the end their quality came through,” he said.
Southampton join Tottenham, Manchester United and the winner of Sunday’s other tie between Leicester City and Chelsea in the semi-finals.