The Gunners needed an own goal from Southampton's Guly Do Prado to equalise Gaston Ramirez's opener as they ended their festive fixture list with a point.
Arsenal scored 15 goals in winning their four Premier League games leading into the clash with the Saints and Gibbs was frustrated not to make that five.
"It was disappointing to break our winning streak because four on the spin left us feeling good and we were so hungry to get three points," he said.
"It was disappointing to only get the draw but we have to move on to the next game.
"(Near the top of the table) is where we expect to be, even more. But the winning run gave us a good lift at this stage of the season because the games come thick and fast at this time of year."
Arsenal boss Arsene Wenger did not feel his side were at the races at St Mary's and Gibbs knows playing well regularly is important.
"Consistency over the whole season, we're looking for that," he said.
"Over this period I've been impressed with everyone and the effort they've put in, credit to the boys.
"But we are looking for more consistency over the whole period of the season."
Gibbs has joined Jack Wilshere, Aaron Ramsey, Alex Oxlade-Chamberlain and Carl Jenkinson in penning new deals at the Emirates Stadium, with Wenger targeting a strong British backbone to his squad.
The 23-year-old England international reckons his new contract and recent form prove his dedication to Arsenal.
"Me and the other boys committed ourselves to the club," he said. "I've always been committed to the club ever since I was given my chance by the boss.
"I've wanted to show him how dedicated I am. I hope the last few months I've done that."
Next up for Arsenal is an FA Cup third-round tie at the Liberty Stadium on Sunday and Gibbs is targeting success in the competition.
"It would be great to get a trophy, of course," he said. "We go to Swansea like it's one of our biggest games of the season. We need to go there and realise that this is another opportunity to get a trophy and we want to go there and win.
"Obviously we're out of one cup now so we need to make sure on Sunday that we're fully focused and realise that we have a chance here to put a trophy in the cabinet."
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