Cook, who was relieved of his One-Day International (ODI) captaincy duties ahead of the quadrennial event, scored his first century for England in more than two years this week in a tour match against Caribbean side St. Kitts and Nevis XI.
Warne said Cook is likely to be a barometer for the team's success.
England are scheduled to face the West Indies and New Zealand before the Ashes series against Australia.
"England need Alastair Cook performing as captain and more importantly as an opening batsman. That's where all the problems started last year with England," Warne was quoted as saying by cricket.com.au on Thursday.
"When he wasn't making runs, that affected everything: the way the team played, their success, and they went to Australia and lost 5-0."
Warne believes Cook making runs will reflect on his captaincy and he will be able to lead his side better.
"For England fans, they want to see Cook back making runs, that will rub off on his captaincy - he'll play well and the team will respond to that too," he said.
"The first innings in that first Test is so crucial to setting the tone for the rest of the series, so fingers crossed for Alastair Cook," he added in reference to the first tour in against the West Indies.
England are expected o beat a weak West Indian side despite not winning a series away from him since toppling India in India in 2012.
"They're playing a depleted West Indies side so they should be able to beat them comfortably," the former leg spinner said.
"And then the Ashes - England always play better in their own conditions so it's definitely going to be more competitive than 5-0. I think England will fancy their chances of winning."
The 45-year-old believes the Ashes this time around will be competitive if the big guns of the England team are fit and ready to fire but added it will still be tough to beat the newly-crowned world champions.
"For me, it's a lot about fitness. The players have to stay fit. Jimmy Anderson and Stuart Broad have to be 100 per cent fit. They had poor World Cups so they'll look forward to getting the red ball back," the spin great said.
"If they perform well, take 20 wickets in the Test match arena, the top order fires with Jonathan Trott and Alastair Cook back, then it's going to be a very competitive Ashes series. But Australia are going to be tough to beat, that's for sure," he concluded.