The benchmark S&P/ASX200 index finished up 55 points, or 0.66 per cent, to 8,402.4, its highest close since December 9.
The broader All Ordinaries gained 58.3 points, or 0.68 per cent, to 8,652.7.
The ASX had been up by as many as 106 points in morning trading but plunged 70 points in 10 minutes just before midday, after Mr Trump said he might impose levies on Canada and Mexico within days.
"We're thinking in terms of 25 per cent (tariffs) on Mexico and Canada, because they're allowing vast numbers of people" into the country, he told reporters from behind the Oval Office's Resolute Desk.
"I think we'll do it February 1," he added.
Those words sent brief shudders through both the ASX and the Aussie dollar, although both had mostly bounced back by late Tuesday afternoon.
Just after 5pm AEDT the Aussie was buying 62.37 US cents, up from 62.06 US cents at Monday's ASX close.
But it had been as high as 62.80 US cents before the Oval Office tariff remarks, its strongest level against the greenback since January 7.
"The first few hours of Trump administration have underscored that the policy environment will be dynamic once again, and markets should brace for volatility," Saxo chief investment strategist Charu Chanana said.
"Clearly, the markets celebrated too soon with tariff threats missing at the outset in Trump's inaugural speech."
But the absence of any threats on China had kept the hopes of negotiation alive, Ms Chanana said.
Six of the ASX's 11 sectors finished higher and four closed lower, with tech basically flat.
Utilities was the biggest loser, dropping 1.6 per cent as Origin Energy retreated 2.3 per cent and AGL fell 1.4 per cent.
The financial sector was the biggest gainer, rising 1.3 per cent amid gains for all four big banks.
Westpac rose 1.4 per cent to $32.74, NAB climbed 2.1 per cent to $38.87, ANZ advanced 2.0 per cent to $30.15 and CBA added 0.7 per cent to $156.20.
The heavyweight mining sector rose 0.9 per cent, with BHP climbing 0.9 per cent to $40.61, Fortescue adding 1.1 per cent to $19.37 and Rio Tinto growing 0.4 per cent to $120.61.
Goldminers also advanced as the precious metal traded near a six-week high of $US2,726 an ounce, up from under $US2,700 last week.
Northern Star rose 1.6 per cent, Evolution gained 2.2 per cent and Regis Resources grew 3.4 per cent.
Lithium developer Liontown Resources was the best performer in the ASX200, rising 11.9 per cent to a seven-week high of 70.5 cents.
Wesfarmers rose 1.6 per cent to $72.88 after the Kmart and Bunnings owner said it would close down its money-losing e-commerce unit Catch.com.au, which Wesfarmers bought for $230 million in 2019 but had struggled against increased competition in Australia's ecommerce industry.
ON THE ASX:
* The benchmark S&P/ASX200 index on Tuesday rose 55 points, or 0.66 per cent, to 8,402.4
* The broader All Ordinaries gained 58.3 points, or 0.68 per cent, at 8,652.7.
CURRENCY SNAPSHOT:
One Australian dollar buys:
* 62.37 US cents, from 62.06 at close of business Monday
* 96.93 Japanese yen, from 96.82 yen
* 60.09 euro cents, from 60.24 euro cents
* 50.80 British pence, from 50.83 pence
* 110.53 NZ cents, from 110.75 NZ cents
