The Ibrox side thumped Tommy Wright's team 5-1 to record their biggest top-flight home win since May 2012.
The triumph also saw them capitalise on Celtic's Rugby Park slip-up as they moved up into second spot.
It completes a swift turnaround for Gers, who found themselves four points adrift of their Old Firm foes after losing the opening derby battle of the campaign at the beginning of the month.
They now lead Brendan Rodgers' side by a point but while former Liverpool left-back Flanagan is not paying too much attention to the Hoops just yet, he does believe his side have shown they are serious contenders this term.
Asked if the result sent out a statement, he said: "I think so, yeah. I think we all knew we had to put on a good performance today and we showed that.
"From the off we were ruthless. The manager said before the game to be ruthless and not to give them a break.
"We can't control what [Celtic] do. We just want to keep picking up points and see where it takes us.
"There was a few updates [from Rugby Park before kick-off] although the game was still going on before we went out for the warm-up.
"Some of us didn't know the final score so to come in after the game and find out they had got beat was great.
"The manager wants us to believe. He has instilled that and has urged us to keep going.
"But it is early days at the minute and we are not focusing on them."
Gers now move on to Wednesday's Betfred Cup quarter-final with Ayr United but Saints have another major task as they take on Celtic in the last eight.
Dejected Saints boss Wright said: "We will be in tomorrow and this is gone. We'll have to try to learn from it.
"Wednesday night gives us an opportunity to get to a semi-final. It's a cup game, a one-off.
"Celtic will be hurting after today but we're hurting as well and we'll be looking for a better performance."