Great Britain are third after the opening dressage stage of the team eventing competition in Greenwich Park.
Heavy rain and lightning threatened to derail Tina Cook's performance but the Briton rode solidly to finish 14th, two places behind team-mate Mary King.
Riding High Kingdom, Zara Phillips, the Princess Royal's daughter, scored 46.10 on her Olympic debut to sit equal 24th.
Favourites Germany lead Australia, with Monday's cross-country stage and Tuesday's showjumping to come.
The three best scores of the five riders in each team decide the medals.
Germany have four riders in the top 11 after the dressage, meaning they could afford to drop world champion Michael Jung's score of 40.60.
King and her mount Imperial Cavalier are the highest-placed Britons in 12th. Cook and Miners Frolic are 14th, William Fox-Pitt (Lionheart) equal 17th, Phillips (High Kingdom) equal 24th and Nicola Wilson (Opposition Buzz) equal 39th.
Phillips, 31, said: "It was incredible to be selected and to be here at home for the Olympics is an amazing feeling."
Double Olympic bronze medallist Cook said: "At the Olympic Games, you hope you are not going to have rain, thunder and lightning thrown at you, but I did and I had to deal with it."
On Monday, riders face the 28-obstacle cross-country course through Greenwich Park. Competitors must complete the course in 10 minutes, three seconds or else they will pick up time penalities.
"We just hope it doesn't rain too much because it will be very slippery around those corners on the cross-country course," Cook added.
"The horses do have studs in their shoes, but if horses are slipping then you will just have to steady around some of the corners, so therefore time faults will rack up."
Japan's Yoshiaki Oiwa leads the individual standings on 38.10, with Italy's Stefano Brecciaroli second on 38.50 with New Zealand legend Mark Todd, the Olympic gold medallist in 1984 and 1988, third on 39.10.