Leicester lost 2-0 at Swansea, taking the champions down to 17th in the table and one point from safety after last year's sensational campaign.
First-half goals from Alfie Mawson and Martin Olsson mean Leicester have claimed just a single point in 2017, but Drinkwater is adamant the players are still behind Ranieri.
He said after the Liberty Stadium clash: "It's down to us on the pitch what happens.
"The manager has our support and we have his support. It's about time we showed that belief and push ourselves a bit more."
Drinkwater, who was crucial in Leicester's title triumph last term, is confident the Foxes can turn things around before the end of the season.
He added: "It's hard to put into words but we need to get out of this and keep on fighting. We need to keep pushing ourselves and make sure we turn this around.
"We're a bunch of honest lads. We work hard and are more disappointed than everyone. We know it's not acceptable and need to get things right quickly.
"We've not changed much, it seems to be we're not getting the rub of the green but we believe in ourselves as a bunch of lads so as long as we keep our heads up, it will come."