MSCI's broadest index of Asia-Pacific shares outside Japan rose 0.2 percent. The index has recovered nearly all the losses suffered last Friday when it plunged in response to Britain's decision to leave the European Union.
Japan's Nikkei rose 0.8 percent after the yen gave back more territory seen during flight-to-quality moves earlier in the week. South Korea's Kospi added 0.3 percent.
The Dow climbed 1.3 percent on Thursday, posting its third straight days of gains and erasing a bulk of its losses after Brexit.
Some of the overnight gains on Wall Street were attributed to hopes for more monetary easing after BOE Governor Mark Carney said on Thursday the central bank would probably need to pump more stimulus into Britain's economy over the summer in wake of the Brexit shock.
A number of Asian central banks, including those in Japan and China, are also expected to ease policy further later this year.
Carney's comments, however, also forced the pound to head south again and cut short the currency's recovery from a 31-year low plumbed on Monday.
Sterling last traded at $1.3341 after nearly touching $1.35 on Thursday. The pound, which lost 8 percent soon after the Brexit vote, had sunk to $1.3122 on Monday, its lowest since 1985.
"Carney has always been nervous about how Brexit would impact the U.K. economy and now that it is a reality, he is preparing for the worst, agrees with the doom and gloom scenario and wants to assure the market that the BoE will be proactive," wrote Kathy Lien, managing director of FX strategy at BK Asset Management.
Sterling's slide added mild pressure on the euro, which was steady at $1.1107 after slipping overnight from $1.1154, it highest in six days.
The dollar held to gains against the safe-haven yen, which has been pulling back on abating risk aversion. The greenback, which fell to 99.00 yen in the initial phases of the Brexit shock, was little changed at 103.115 yen following mild gains overnight.
In commodities, crude oil inched up in early trade after sliding overnight on returning Nigerian and Canadian crude output from outages and as traders booked profits at the end of the best quarter in seven years.
U.S. crude was up 0.2 percent at $48.42 a barrel after shedding 3 percent on Thursday.