Arsenal are not far behind, but Gibbs believes the Saints' challenge is here to stay for this season at least.
He wrote: "The race for the top four places is just as competitive as ever, and Southampton are proving they aren’t going anywhere. I’m not surprised to see them where they are, they are having a great season, and it’s a similar situation to what we saw with Everton in recent years.
"Southampton are having a great run, and probably don’t feel any kind of pressure because they aren’t expected to be up there. It’s going to be very interesting in the months to come and it goes without saying we need to be at our best week in week out."
His suggestion Southampton don't feel any pressure hints at a possible advantage he feels Ronald Koeman's side have, compared to the expectation at the Emirates, with this group of players desperate not to be the group which lets Arsene Wenger down as the first he has managed to lead into the top four places.
The win away at Manchester City was a great sign, but Southampton do not either have European competition to worry about as a distraction, unlike Arsenal.
Southampton appear to have got their wobble in terms of form out of the way a couple of months back, and their ability to pick up points against the so-called top sides gives them a real confidence boost and shows opponents like Gibbs they are serious in their challenge.