Crime in England and Wales fell by almost a third in the first two full months of lockdown, according to the Office for National Statistics.
This was driven by falls in reported thefts and burglary, the ONS said.
But the ONS said drug crime rose by up to 44% compared with the same period last year, due to targeted policing.
A survey also suggests the vast majority of adults (91%) are satisfied with the way police have dealt with the coronavirus restrictions.
The figures, which are based on telephone research, support earlier data from police forces of a significant decline in offending during April and May.
Stop and searches
The ONS data shows drug offences recorded by police rose by 22% in April to 16,570 and 44% in May to 20,687, compared with 13,535 and 14,343 for the respective months in 2019.
The ONS report linked this to "proactive police activity in pursuing these crimes during lockdown".
It said the rises were driven largely by drugs possession offences with "early indications" suggesting this was "particularly evident" in London, where the Metropolitan Police had increased the number of drugs-related stop and searches it was carrying out during that time.
Meanwhile, there was a 57% rise in computer misuse offences compared with the two-month average from July to December 2019.
The ONS said the increase was not "statistically significant" because of the small sample size and it will be explored further.
Overall police-recorded crime during the coronavirus lockdown was 25% lower in April and 20% lower in May compared with the same period in 2019.
It also fell 5% in March compared with February, the research found.
Reports of theft fell in April and May to "almost half the level recorded" during those months in the previous year, the ONS said.
But reports of crime rose again as lockdown restrictions began to ease.
Billy Gazard, from the ONS Centre for Crime and Justice, said the "significant" fall in crime at the height of the coronavirus lockdown in England and Wales "was driven by reductions in theft offences, particularly domestic burglary and theft of personal property".
He added that the findings were "not unexpected" as the period coincided with the majority of people spending long periods at home.
But he said the exception was police recording of drug offences, adding: "This reflects proactive police activity as overall crime levels reduced."
The report gives the first official indication of some police-recorded crime figures since the pandemic took hold in the UK.
However, its findings are limited due to the difficulties in gathering statistics posed by the circumstances caused by the pandemic and because some figures are not yet available.
For example, police reports of domestic abuse are recorded quarterly, so official figures indicating the prevalence of this crime during lockdown have not yet been made public.