Wall Street and Chinese equities and emerging market currencies were all boosted by Trump's decision, which alleviated investor concerns about pressure on corporate results in the sector.
"In general it has been a risk-on day," said Greg Anderson, global head of foreign exchange strategy at BMO Capital Markets.
Trump added Huawei to a trade blacklist last week, leading several companies to suspend business with the world's largest telecoms equipment maker, which could weigh on their sales. Chipmakers, many of which sell to Huawei, bore the brunt of Monday's sell-off. But late on Monday, the United States granted Huawei a license to buy U.S. goods until Aug. 19.
Some analysts, however, warned that the relief could be temporary.
"Asian equity markets reacted positively to the news, but there should only be limited follow-through, as uncertainty remains high, and this uncertainty may potentially prompt companies to cut back on existing capex plans," wrote Hans Redeker, global head of foreign exchange strategy at Morgan Stanley.
Against the Japanese yen, the U.S. dollar was half a percent stronger, last at 110.59 yen. The dollar index which measures the greenback against a basket of six rivals, was 0.12% higher, last at 98.051.
Against the euro, the dollar was up 0.1%, last at $1.116. The single currency is being hurt by dollar strength and by the upcoming European parliamentary elections in which euroskeptic parties may fare well.
The dollar was also stronger against the pound, which boomeranged on developments in Britain's plan to leave the European Union.
Prime Minister Theresa May set out a "new deal" on Tuesday for Britain's departure from the EU, offering sweeteners to Parliament including the chance to vote on whether to hold a second referendum to try to break the impasse over Brexit. The pound initially jumped on the news, then gave up the gains after it became clear that Parliament - which thus far has opposed a second public vote - would have to back any new referendum.
"I don't think there's any way that passes through parliament," said Anderson. "Unless something changes dramatically, the votes aren't there."
(Reporting by Kate Duguid; Additional reporting by Abhinav Ramnarayan; editing by Jonathan Oatis and Leslie Adler)
By Kate Duguid