Goals from Casemiro and substitute Gareth Bale – his 100th for the club – either side of a coolly converted penalty from captain Sergio Ramos moved Madrid a point ahead of their neighbours and up to second in the standings.
Barcelona are still five points better off and can extend their lead further when they travel to face Athletic Bilbao on Sunday.
However, with Madrid resurgent and a Clasico clash at Santiago Bernabeu coming up before the end of the month, Solari suggested he smells blood.
"Real Madrid fight for everything until the end - it's not over, it's historic," he told reporters.
"Now we are second, but we have to keep going because I know that Atletico will keep fighting.
"They're a great team with a marvellous coach, now we have to keep this up and keep winning. We have our feet on the ground and know that we have to keep working."
Despite Bale bringing up an impressive personal milestone, Solari had no doubt his team's victory in El Derbi was forged by the work of the collective.
"It was a team victory, there's no other way to get anything from this stadium against a strong and competitive rival," he added.
"Playing as a team is the only way to win at the Wanda, we were strong in every aspect of play. It's the best we've played since December."