The Light Blues found themselves under fresh pressure on Saturday as they fell 1-0 down to Motherwell just days after being shocked by St Mirren in the Betfred Cup.
But the Ibrox men kept their cool and hit back with a Kemar Roofe double either side of Cedric Itten's header to finally kill off Stephen Robinson's ultra-defensive Motherwell.
Barisic, speaking ahead of Wednesday's trip to St Johnstone, said: "I think that we showed character because I remember last season and the season before when we would be 1-0 down, we'd always struggle to come back.
"We would start to panic and maybe we couldn't come back.
"But this time we believed in each other on the pitch, didn't panic and in the end I think we deserved that win. We again scored three goals and showed the team spirit we have and a very different character from last season and the year before."
That win against Well opened up a 16-point lead over Celtic at the top of the table.
But while Barisic shrugged off talk that his side are now showing title-winning credentials, he did praise the speed at which Gerrard's new recruits this season have grasped the enormity of what is at stake as they chase down the club's first major trophy in a decade.
He said: "I cannot talk about the mentality of champions but yes I can say that the mentality is different.
"This is now our third season together and we've learned a lot about everyone and each other.
"There are some new players who have adapted very quickly.
"I think our mentality now is much different and I can see that especially in this last game (against Motherwell). I can't remember a game (previously) where we were 1-0 down and that we didn't panic and kept playing to our game plan so I think there is a much different mentality from our squad, which is very good and I think our results show that."