Nino Serdarusic gave Croatia an opening match win over Hungary to secure first place in Group D and a place in the last eight of the Davis Cup against Italy.
His 6-4, 6-4 success was the deciding factor as Marin Cilic lost the second singles 4-6, 7-5, 6-4 to youngster Zsombor Piros.
Kazakhstan also moved into the knockout stage by beating 2019 runners-up Canada in Madrid.
The Kazaks, who beat Sweden on Saturday, saw Mikhail Kukushkin and Alexander Bublik, beat Brayden Schnur and Vasek Pospisil.
Britain got off to a poor start against the Czech Republic as they looked to secure top spot in Group C.
Dan Evans lost the first singles rubber against Tomas Machac, before Cameron Norrie levelled the tie and doubles pair Joe Salisbury and Neal Skupski won in straight sets to take out the tie 2-1 , beating France to top spot.