Oil prices hit multiyear highs above $70 a barrel after OPEC and its allies forecast higher demand and boosted output—punctuating a global economic reawakening that has raised the prices of a broad range of commodities.

The move by a group of oil producers led by Saudi Arabia and Russia amounted to a continued unwinding of steep cuts they made at the start of the pandemic. Recent oil-price milestones come alongside similar ones hit by commodities from tin and copper to lumber. They have all soared too amid pent-up demand that has producers of the stuff struggling to keep up.

“It is a good, old-fashioned reflation trade,” said Tom Price, head of commodities strategy at investment bank Liberum, describing expectations of a burst of economic activity that boosts assets. Mr. Price said such a broad grouping of commodities hasn’t risen simultaneously for so long since the economic recovery that followed the 2008-09 financial crisis.

Brent crude, the international energy benchmark, rose 1.3% to $70.25 a barrel, notching its highest close since May 2019. West Texas Intermediate futures, the key U.S. gauge, gained 2.1% to $67.72 a barrel to close at its highest level since June 2018.

Members of the Organization of the Petroleum Exporting Countries and their allies, a group known as OPEC+, agreed Tuesday to a previously planned output increase of about 450,000 barrels a day, starting next month. Saudi Arabia, meanwhile, agreed to continue easing separate, unilateral cuts of one million barrels a day that it put in place earlier this year.

In April, the group agreed to increase output by more than two million barrels a day by the end of July, bringing cumulative additions over the past year to some four million barrels a day. That is a big chunk of the 9.7 million barrels a day the group agreed to cut early in 2020 when the coronavirus first started shutting down economies, sapping global crude demand and sinking prices.

Now, with infection rates generally in check in much of Asia and China—the world’s biggest oil consumer—and vaccine drives in the U.S. and Europe plowing ahead, OPEC and a group of non-OPEC producers led by Russia are betting markets are ready again for more oil. A technical committee of the OPEC+ group forecast on Monday that oil demand would jump by six million barrels a day in the second half, according to OPEC delegates. As a result, global oil stocks will fall below their five-year average for the 2015-2019 period by the end of July, signaling an end to the pandemic glut, they predicted.

Demand for oil and other commodities has mirrored the huge swings in global economic activity during the pandemic. Having fallen sharply as large parts of the global economy were placed in suspended animation during the second quarter of last year, demand for oil has rebounded as many rich economies have thawed over recent months.

Strong demand for an array of goods that helped households and businesses adapt to the constraints of living with the Covid-19 virus had pushed global industrial production above its pre-pandemic peak by the end of last year. Growth since then has left the world’s factories hungry for energy and raw materials as a result. A weaker dollar is also making dollar-priced commodities more attractive.

Demand is set to rise further this year as the global economy is forecast to record one of its most rapid expansions in many decades. The Organization for Economic Cooperation and Development said Monday it expects global output to increase by 5.8%, which would be the strongest expansion since 1973.

Soaring demand for commodities to feed that expansion could act as a break if supply can’t keep up. Higher oil prices have contributed to a pickup in inflation around the world. Figures for the eurozone released Tuesday showed consumer prices were 2% higher in May than a year earlier, the fastest rise since late 2018. But much of that rise was down to energy prices, which were 13.1% higher than a year earlier.

In the U.S., the Commerce Department’s inflation measure showed consumer prices rose 0.6% in April from a month earlier and 3.6% from a year earlier. Core prices, which exclude energy and food, rose 0.7% over the month and 3.1% over the year.

As a large commodities importer and the first major economy to bounce back from Covid-19, China has had to absorb higher costs for a range of commodities from crude oil to copper and iron ore since last year. That has created new challenges for China’s economy.

Especially affected are smaller manufacturers, which have been suffering from thinner profit margins in recent months because of high raw-materials prices. In response to those rising cost pressures, more Chinese factories have raised product prices recently. Some others have halted operations temporarily and turned down new orders. The country’s factory-gate prices jumped by 6.8% year over year in April, the most in 3½ years, driven by surging commodities prices.

Central bankers have said they expect inflationary pressures to ease toward year-end as commodity producers and factories respond to higher prices by raising production. OPEC’s move Tuesday underpinned that view.

“We have rising oil and commodity prices, and we also have some weird effects because of changes in the pattern of consumption,” said Laurence Boone, the OECD’s chief economist. “But this should fade as the supply response kicks in.”

Still, rising prices are already making life tougher for manufacturers. Companies as diverse as General Motors Co. and Vestas Wind Systems A/S, a Danish maker of wind turbines, have complained about the rising price of steel.

“It’s a big issue for car makers,” said Andy Palmer, who has been in the industry for 40 years, including as the former chief executive of British auto maker Aston Martin. “I have seen periods of time where commodity prices have been volatile, but it’s rare to see almost all raw materials rising so quickly,” he said.

While the global oil spigot can be turned on relatively quickly—OPEC can draw from its prodigious untapped capacity—it is much harder for miners and farmers to suddenly increase production of coal, copper or cotton. The price gains are pushing mining companies, for instance, to examine their future plans.

Commodity trading company Glencore PLC said in February it was looking at restarting production at the world’s largest cobalt mine in the Democratic Republic of Congo. Glencore opted to close the Mutanda mine, which also produces significant amounts of copper, in August 2019, as prices for both metals tumbled. In recent months, lingering supply bottlenecks caused by the pandemic and booming demand from rebounding Asian economies have sent prices soaring. Copper prices surged to their highest-ever level last month. Cobalt prices have risen over 50% this year.

—Summer Said in Dubai, Stella Yifan Xie in Hong Kong and Paul Hannon and William Horner in London contributed to this article.

Write to Benoit Faucon at benoit.faucon@wsj.com and Alistair MacDonald at alistair.macdonald@wsj.com