Canadian second seed Raonic sent down 16 aces on his way to a 7-6 (7-2) 6-2 win over American Steve Johnson in their quarter-final in Washington.
Raonic converted two break points while giving up no opportunities as his big serve gave Johnson little chance.
Neither player was able to edge ahead in the first set before Raonic steamed 5-1 ahead in the tie-break on his way to taking the first.
The world number seven broke in the opening game of the second set and again in the seventh as he eased through at the ATP 500 event played on hard courts.
Raonic will meet American left-hander Donald Young in the semi-finals after the world number 73 upset South African seventh seed Kevin Anderson 3-6 7-6 (7-3) 6-2.
Young broke serve twice in the deciding set on his way to the win.
French sixth seed Gasquet took just 59 minutes to thrash Japanese fourth seed Kei Nishikori 6-1 6-4.
Gasquet served nine aces and broke three times as Nishikori never looked likely.
The 28-year-old will have to wait until Saturday to find out his semi-final opponent.
The clash between Vasek Pospisil and Colombian 10th seed Santiago Giraldo was suspended due to rain.
Pospisil, the Canadian 13th seed, claimed the second set 6-3 after losing the first 7-6 (7-4).