The extra-time triumph sets up a Euros quarterfinal against Switzerland.
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"We've got a togetherness, we proved that tonight," Rice told BBC Sport. "We'd do anything to protect this manager, protect each other.
"We'll keep going and keep fighting and what we've got as a group tonight showed out there. It's an honour to be a part of it and we're going to give it everything."
Jude Bellingham, of Real Madrid, sent the game into extra-time.
"We've won this game together," the 21-year-old told ITV Sport. "Not me, not Harry Kane, not the individual moments.
"It's the likes of Ivan Toney, Eberechi Eze, Cole Palmer, Bukayo Saka going to left-back.
"It's the sacrifice you make for the team and that's the energy we need to keep regardless of what goes on."