The Ibrox skipper was frustrated to see the hard work Gers put in to beat Celtic before the winter break go to waste as they froze on their return to action.
Jordan Jones - who will join the Light Blues this summer - showed his loyalties still lie with Kilmarnock for now as he fired Wednesday night’s Rugby Park winner.
The 2-1 defeat means Steven Gerrard’s side drop back down to third place, three points behind their bitter Parkhead rivals who eased past St Mirren 4-0.
And with Tavernier predicting the Hoops will continue racking up the wins, he confessed his side will not get away with another slip-up.
“We’ve been in the same position before when we have dropped points, but we can’t afford to drop any more,” said the Rangers captain.
“We’ve got to expect Celtic to win every game, so what we can do is win every game going forward.
“We have to put in a lot better performance against Livingston at the weekend and repay all our fans with a performance that they deserve.
“It was very disappointing at Kilmarnock. We made a good start, our formation was working really and we got the opening goal. But in this league, if you are going to gift teams chances as we did with two, they are going to punish you.
“It’s so disappointing for the boys, especially given that three weeks ago we put in the type of performance that we did against Celtic.
“I wish we could have replicated that, but it wasn’t there after the 20 minutes and when we conceded the goal. We need to have a long hard look at ourselves.”
Jermain Defoe took just 12 minutes to open his Rangers account but the former England striker’s debut goal was pegged back soon after as Joe Worrall’s blunder handed Eamonn Brophy the equaliser.
Jones then pounced on a misplaced Ryan Kent pass to hand Steve Clarke’s side victory.
The Ibrox faithful have been licking their lips in anticipation of seeing Defoe in action but Tavernier admits the 36-year-old veteran can forget about viewing the Premiership as a cosy place to wind down his career.
He said: “It’s a tough league. Everyone we play wants to beat us. They give 120 per cent against us and we expect that.
“But the only team that is going to hurt us is ourselves - that’s what we did tonight.
“Jermain is an experienced player, so he’ll know how to adapt to everything. He will prove to be a good signing as he showed by scoring.
“Joe has already held his hand up for his mistake and we also said before the game that, on the surface, square passes might get intercepted and might cost us - that’s what happened with the second.
“Still (Jones) was able to go a long way along the ground and then put it into the net.
“That was their two main chances and they tucked them away, but it was criminal from us to gift those chances away.”