Haase, who upset seventh-seeded Grigor Dimitrov in the third round, was in his first career semifinal of a Masters Series tournament.
The 36-year-old began the second set with a break and cruised through the rest of the match, which lasted just over an hour, to book a last-four clash with Dutchman Robin Haase.
The 18-year-old from Richmond Hill, Ont., was coming off a second-round win over 2009 U.S. Open champion Juan Martin Del Potro and a thrilling three-set upset of top-seeded tennis legend Rafael Nadal in the Round of 16.
Holland's world number 52 Haase fought back after losing the first set against Argentina's Diego Schwartzman to win 4-6, 6-3, 6-3.
Next up for Federer in Friday's quarter-finals of this US Open tune-up event will be 12th-seeded Spaniard Roberto Bautista-Agut, who saved a match point in a narrow 4-6, 7-6 (7-5), 7-6 (7-2) victory on Thursday over unseeded Frenchman Gael Monfils.
Federer, who is seeking his third Rogers Cup title after triumphs in Toronto in 2004 and 2006, will face either German Alexander Zverev or Canadian Denis Shapovalov in Sunday's final.
N. Korea threats will be met with 'fire and fury'
He urged North Korea to first halt tests for an "extended period", however often such confidence-building measures have failed. The question of whether Beijing would go along with much harsher sanctions against North Korea was answered this weekend.
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Trump reciprocated the sentiment and suggested that the stalemate between the USA and North Korea could have had a different outcome, had he been the president in years past.
"I'm happy I'm playing as well as I am this year on the match courts. I got some decent rhythm going".
Federer moved 4-1 ahead after securing a couple of mini-breaks, only for Haase to respond again by taking the next two points against serve and then moved 5-4 ahead.
"It's tough to come out and serve after a 12-minute rain delay".
"I'm looking forward to a tough match because he can serve very well and he mixes up his tactics a lot", Federer said. "But, as I have practiced with him quite a bit, maybe I am better prepared than if I would have never hit with him before".
Haase rallied after losing the opening set to improve to 5-0 all-time against Schwartzman.