The Foxes' 3-1 win over Manchester City on Saturday saw Claudio Ranieri's men go five points clear of United's cross-city rivals with 13 games still to play.
United, currently occupying fifth place, take on defending champions Chelsea on Sunday at Stamford Bridge and Carrick believes a positive result could be the catalyst for a possible league triumph.
"Sunday's a big game for us, of course it is. If we can go there and get the win, it puts us in a good position," he told Sky Sports.
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"But then we have to go and do it all over again the week after. It's week by week now and we are coming to the time of year where it's almost the business end where things are won and lost.
"We still want to be up there challenging and fighting and giving ourselves every opportunity to still finish on top.
"I've been here for a while now and there's been seasons where it hasn't gone so well up until now and you find yourself putting a run together, winning games and you end up having a successful season.
"That's the focus from now, really. This season almost starts now for us - putting a run together and hopefully ending up with some silverware again.
"Can we win the title? I think we can. I believe we can. I'm realistic to understand where we are and the situation, but as games tick by, if we keep winning then we give ourselves a chance.
"If we don't keep winning, then the chance falls by the wayside. It's up to us carry on the form we've showed over the last couple of games and build on that."