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.
"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."
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.
"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," he added.
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."