The new restrictions that took effect at midnight limit entry from those areas to German citizens and residents, truck drivers, transport and health service staff and a few others, who have to register online and show a negative coronavirus test.
Interior Minister Horst Seehofer told the Bild am Sonntag newspaper that the new checks may cause some delays at the border and German police "will not just wave traffic through."
Infection rates in Germany have been declining steadily in recent weeks but officials are concerned about the possible impact of variants first discovered in Britain and South Africa. Both variants have been reported in Germany but so far appear to account for just a small proportion of cases.
Significant numbers of cases of the latter have been discovered in Tyrol, whose residents have needed to show a recent negative coronavirus test to travel to the rest of Austria since Friday. The spread of the British variant has prompted a complete lockdown of some Czech districts on the border with Germany and Poland.
To prevent a pile-up of trucks inside Austria, regional officials in Tyrol planned to check trucks headed for Germany at the Brenner pass crossing with Italy to ensure that drivers have the necessary paperwork to enter Germany.
Chancellor Angela Merkel and Germany's 16 state governors agreed on Wednesday to extend most of the country's lockdown restrictions until March 7, though schools and hairdressers can open sooner.
They set a new target of 35 new cases per 100,000 inhabitants per week before letting small stores, museums and other businesses reopen. That figure stood at 57.4 on Sunday, down from a peak of nearly 200 just before Christmas.
The eastern state of Saxony's governor cautioned Germans against expecting too much too soon.
"Unfortunately, there can't be Easter vacations in Germany this year," Michael Kretschmer was quoted as telling Bild am Sonntag. "Too much mobility as a result of travel and tourism already in April would be poison. We would destroy everything we have achieved since mid-December."
Hotels and restaurants in Saxony, which was hard hit in the fall and winter, will have to remain shut over Easter, and the reopening of theaters and operas will have to wait until after Easter, he said.