Eaton, who set the world record of 9,039 points at the US Trials in June, made an excellent start with an Olympic Decathlon best of 10.35 seconds in the 100m, before following that with a leap of 8.03m in the Long Jump.
Hardee, who beat Eaton to the world title in Daegu last year, reduced the deficit with 15.28m in the shot to Eaton's 14.66m, but 24-year-old Eaton remains the favourite to claim gold when the competition concludes tomorrow.
World champion Pawel Wojciechowski was a shock casualty in the men's Pole Vault, the 23-year-old Pole failing with all three attempts at his opening height of 5.35m to crash out.
France's European champion Renaud Lavillenie and Germany's Raphael Holzdeppe led the 14 qualifiers for Friday's final with clearances of 5.65m.
Poland's Anita Wlodarczyk led the qualifiers for the women's Hammer final with a throw of 75.68m on her 27th birthday, with Britain's Sophie Hitchon delighting a capacity home crowd with a national record of 71.98m to advance.
Team-mate Mo Farah remained on course for the long-distance double following his 10,000m triumph on Saturday, although the 29-year-old had to work hard to finish third in his heat of the men's 5000m.
Ethiopia's Dejen Gebremeskel won the much quicker second heat to lead the qualifiers for Saturday's final.
Former world champion Caster Semenya safely advanced to the semi-finals of the women's 800m, an event which also saw two performances where times were irrelevant.
California-based 19-year-old Sarah Attar became the first Saudi Arabian woman to compete in Athletics at the Olympic Games when she finished last in the final heat in 2:44.95, while Turkey's Merve Aydin limped around in 3:24.35 after suffering an injury just a few metres into heat two but showing determination to finish as she refused to step off the track.