The Blues forked out a club-record £97.5m to re-sign the 28-year-old from Inter Milan, eight years after he left the club permanently to join Everton.
Lukaku has made a bright start to life at Stamford Bridge, scoring three goals in as many Premier League appearances, and Cole believes his acquisition can help the Blues retain their European crown.
"Before the season, everyone knew Chelsea were in the market for a number nine and realistically, there were only five names who were going to cut it. You are looking at Robert Lewandowski, Erling Braut Haaland, Kylian Mbappe, Cristiano Ronaldo and Lukaku, all of them in the very top bracket," Cole told BT Sport.
"None of them comes cheap because they are the best in the business, but when you break it down, and look at the attributes Chelsea needed, Lukaku was a great piece of business. We're talking about a proven goalscorer, who is 28 and at the peak of his powers.
"This is the right time for him to take on the responsibility, and he will give Chelsea that extra four or five percent which could help them win the Champions League again."
Lukaku, who is Belgium's all-time leading scorer with 67 strikes to his name, found the net 47 times across two Serie A seasons with the Italian outfit, winning the Scudetto last year. body check tags ::