More than 2,000 Newcastle fans made the trip to St Mary's but saw a less than inspiring display as the Saints claimed a 2-0 win.
"We can't apologise enough," Ameobi told Newcastle's site.
"It was an unacceptable display and if I had travelled all the way down to watch that I would be devastated.
"Hopefully we can get back on the right track at Stoke on Wednesday night and give them something to cheer about."
Ameobi admitted that they could offer few excuses for the display.
"Everyone is so disappointed, we weren't good enough and that is hard to take," he said.
"We've got so many injuries and had another really busy week as well with the Europa League.
"But that is not an excuse for the performance we put in out there today. We were too sloppy at times.
"We are slowly creeping down the table and we have to start getting results as soon as possible to get us back towards where we want, and need, to be."