Brent crude was up $1.06, or 1.9 per cent, at $56.85 per barrel by 1616 GMT (9:46 pm in India). US crude rose $1.42, or 2.9 per cent, to $51.00.
From a technical standpoint, US crude has staged an impressive rebound from the $49.08 level and a decisive breakout above $51.00 should encourage a further incline towards $52.40 a barrel, said Lukman Otunuga, research analyst at FXTM.
"In an alternative scenario, sustained weakness below $49.00, which is also under the 50 daily simple moving average, may open a path towards $47.80."
OPEC, Russia and other non-member producers are cutting output by about 1.8 million barrels per day (bpd) until next March to get rid of a price-sapping supply glut.
The group is increasingly confident that the market is rebalancing fast, helped by the cutback as well as by stronger-than-expected growth in global demand.
The deal is working, keeping oil prices within "a reasonable range", the RIA news agency cited Russian Prime Minister Dmitry Medvedev as saying.
The chief executive of trading firm Gunvor, Torbjorn Tornqvist, also said the market was rebalancing, citing falling product stocks and crude held in floating storage clearing up.
"We don't see this market being out of balance one way or another," he told the Reuters Global Commodities Summit taking place this week. Overall crude stocks "are still high", he added, and OPEC needed to stick to its output curbs.
Short-term price support was coming from the United States, where 85 per cent of US Gulf of Mexico oil production, or 1.49 million bpd, was offline following Hurricane Nate, according to official figures.
Long term, US oil output could be set for a last spike in 2018 before growth flattens for a number of years as rising costs make a big chunk of production uneconomic, the head of top oil trader Vitol, Ian Taylor, told Reuters.
OPEC has managed record-high adherence to its supply cutting deal this year and is considering extending the deal beyond its March 2018 expiry. Some analysts have been concerned that a price recovery could tempt producers to open the taps again.
But analysts at JP Morgan said this was less of an issue, saying "concerns that OPEC compliance would fade into the fourth quarter now appear unfounded."
"Stronger-than-assumed economic growth offers the potential for tight market conditions to continue if OPEC extends the current deal for another nine months," the bank said.