The 30-stock index recovered 123.74 points to 44,860.31.
The much-broader index gained 34.26 points to 6,021.63.
The NASDAQ Composite picked up 119.46 points to 19,174.30.
Both the Dow and S&P 500 notched new intraday and closing records.
Trump on Monday night called for a 25% tariff on products from Mexico and Canada, as well as an additional 10% levy on Chinese goods.
He has already said he would impose a tariff of up to 20% on all imports, and an additional duty of at least 60% on products from China.
To be sure, Wall Street appeared to be taking the news in stride.
While stocks were mixed, some individual names and funds moved as investors analyzed potential impacts from the policies Trump discussed. Automakers Ford gave back 2.6% and General Motors slid more than 9%. Alcohol company Constellation Brands, known in part for Mexican beers Corona and Modelo, fell more than 3%.
Investors also analyzed minutes from the Federal Reserve meeting released Tuesday afternoon. The central bank said it expects interest rate cuts in the future, but to only expect them “gradually.”
Prices for the 10-year Treasury slid, raising yields to 4.30% from Monday’s 4.26%. Treasury prices and yields move in opposite directions.
Oil prices faded 30 cents to $68.64 U.S. a barrel.
Prices for gold perked $13.70 an ounce to $2,632.20 U.S.