GB lead on 113.90, ahead of Germany (123.00) and Sweden (128.50) in Strzegom, going into Sunday's showjumping finale in Poland.
British rider Nicola Wilson, aboard Bulana, is fourth individually behind the German 1-2 of Ingrid Klimke and Michael Jung after the cross-country.
GB's Gemma Tattersall is fifth, with Tina Cook sixth and Ros Canter eighth.
Cook, who won individual European gold eight years ago, delivered a stunning clear round inside the time on Billy the Red.
Her fellow London 2012 team silver medallist Wilson collected just 0.4 of a time fault and Canter 1.6 aboard Allstar B.
It meant that Britain could discount Oliver Townend's score - he collected 40 jumping penalties on Cooley SRS.
"The team needed a clear round, and I am delighted that we delivered that," said Cook.
Britain, whose new coaching chief is former Germany team boss Chris Bartle, can afford to have two showjumping fences down between their team members and still win, even if the German riders all go clear.
German Bettina Hoy, the leader after the dressage stage, was eliminated in the cross-country after a fall, while France are out of the team competition after two of their riders fell.
Leading individual positions after cross-country: 1 Ingrid Klimke (Ger) (Horse: Horseware Hale Bob OLD) 30.30, 2 Michael Jung (Ger) (fischerRocana FST) 32.80, 3 Sara Algotsson Ostholt (Swe) (Reality 39) 35.00, 4 Nicola Watson (GB) (Bulana) 35.50, 5 Gemma Tattersall (GB) (Quicklook V) 36.70
Team standings:1 Great Britain 113.90, 2 Germany 123.00, 3 Sweden 128.50, 4 Italy 177.50, 5 Belgium 196.60
Great Britain's eventers are in pole position to win a first European Championships team gold since 2009.