The shortages started last week and peaked this weekend triggering protests and relentless criticism of the government as TV footages showed hundreds of people with cars and motorbikes lining at stations to get petrol.
Abbasi addressed the media after Sharif chaired a high- level meeting to resolve the crisis.
The minister took responsibility for the shortages and promised a probe to punish the culprits.
He said that the major cause of shortage was the increase in demand following more than 20 per cent decrease in the oil prices in the last two months.
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Abbasi said the artificial shortage will be resolved within a week and announced to further reduce the oil prices in February.
Earlier, Sharif vented his anger during the meeting, attended by key ministers and officials.
"The Prime Minister showed his extreme displeasure over the difficulties being faced by the people due to non-availability of fuel," according to a statement released by the PM House.
Sharif was informed that measures are being taken to provide immediate relief to the people as 15,600 tons of petrol was supplied today all over the country against an average daily consumption of 12,000 tons.
Talking about power situation, the Prime Minister directed the Ministry of Water and Power to immediately work to reduce line losses and recover outstanding amount of over Rs 500 billion in order to clear the amounts due to Pakistan State Oil company and other liabilities.
Separately, finance minister Ishaq Dar termed the petrol crisis as a "conspiracy to defame the government".
He said those responsible will be bought to book.
The crisis started when Sharif was visiting Saudi Arabia, prompting the hashtag #NawazPetrolDonoGhayab - where critics flayed the government for incompetency.