TORONTO — Dominating Tampa Bay’s anemic offense is one thing. Holding the best lineup in baseball, the Toronto Blue Jays, scoreless for five consecutive innings after spotting them a 3-0 lead is quite another.

Rich Hill is impressing the world right now.

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The 35-year-old lefthander has become one of the feel-good late-season stories for the Red Sox. He went up against a goliath, the monster-hitting Blue Jays. Oh, Dioner Navarro got him for a two-run homer in the second inning when Hill allowed all three of his runs. But from there, Hill was almost unhittable.

He stuck with that dramatic curveball he threw at different speeds and different angles, and his fastball, coming in at 91-92 miles per hour, looked even sharper. And he reintroduced his changeup, which he resurrected between starts.

Hill has learned something about pitching all these years in the game, as a starter six years ago and as a reliever.

He will become a free agent after the season, and you can bet a team already is thinking about a low-cost starter it can insert at the back end of the rotation. The scouts on hand from Houston, Kansas City, and Philadelphia liked what they saw.

The Red Sox could try to keep him, but there doesn’t appear to be room in their rotation, especially if they land an ace, and it doesn’t appear Hill wants to be a reliever again after he’s worked so hard at coming back as a starter.

Anyway, why would you want a guy relieving who over two starts has pitched 14 innings, struck out 20, walked one, and allowed three runs on eight hits? He did it vs. American League East competition, vs. one of the worst lineups and one of the best lineups in baseball.

Navarro also singled off Hill in the fourth and was really the only Blue Jays batter who had a clue. Ben Revere did knock in the third run off Hill in the second inning, but Revere was tormented in his other at-bats with two strikeouts and a pop to center.

The dangerous Jose Bautista managed a single to left field in the fifth. The other Danger Brother, Edwin Encarnacion, singled to lead off that second inning, but he also struck out twice.

The curveball remains Hill’s killer pitch. There’s so much movement, so much sweeping. Some 12-6, but sometimes east-west. He’s got a cutter that looks like a curve.

So Hill came out of this very good outing having thrown 107 pitches over seven innings. He allowed seven hits, three runs, walked one, and struck out 10 — for the second consecutive start. He’s the first Sox pitcher to strike out 10 in consecutive games since Felix Doubront in September 2012. He’s the first Red Sox pitcher to do it in his first two starts with the team.

You can point out that Toronto doesn’t have a lot of scouting or data on the new Hill. But Hill stepped up to the mound during a very important time for the first-place Blue Jays, and he allowed the Red Sox to take two out of three games.

Entering this game there was banter as to how effective Hill could be against this vaunted lineup. The Jays usually crush lefthanded pitching (now 20-12 vs. lefty starters), but they couldn’t touch Hill after the second inning.

Hill got the nod as Boston’s sixth starter as the Sox try to reduce innings for youngsters Eduardo Rodriguez and Henry Owens. The Red Sox did not expect anything like this. The Sox inserted Hill and hoped he would give them five good innings.

Now he’s done things that starting pitchers haven’t done in years. Wade Miley, Clay Buchholz, Rick Porcello, and Joe Kelly have not done what Hill has done in two consecutive starts.

Certainly we have to come to grips with what this is. It’s September, and baseball guys don’t like to evaluate any player in September. Yet Hill has just pitched well against the league’s hottest team; the Jays are 40-18 since the All-Star break.

And the Jays obviously have plenty at stake, but Hill kept them at bay for five consecutive innings in which he gave the Sox’ offense a chance to reboot and surge ahead.

Hill, who in early August was pitching for the Long Island Ducks and before that was working out with Milton’s American Legion team, is crafting quite a story.

Although he claimed he had “similar stuff” his last outing, Hill said “mixing in the changeup in the fourth inning” was really a key. Hill said he threw it earlier this year with Syracuse, the Nationals’ Triple A team, then worked on it between starts. He wound up giving Jays hitters not only the curveball look, but the changeup came at a different angle that kept Blue Jays hitters off balance.

Hill talks about “staying in the moment” and concentrating on every pitch. He wasn’t surprised by the Navarro homer because “the pitch was up and he took advantage of that.” But even as the second inning got a bit away from him he never felt out of control.

“Taking that pitch-to-pitch approach, Navarro did a good job,’’ Hill said. “I moved on to the next pitch. I looked forward to making another quality pitch after that.”

But two 10-strikeout games?

Hill doesn’t put too much stock in it. Nor does he revel in earning his first win.

“Initially I just wanted to go out there and start again,’’ he said. “You put together consecutive starts and pitches, then it became more of a reality.

“I definitely wanted to go in this direction, but the results are irrelevant. It’s all about the process. If the end result is a strikeout, then that’s the result. It’s all about staying in that moment.”

He impressed interim manager Torey Lovullo. He impressed president of baseball operations Dave Dombrowski.

Two important people to impress.

He’s the September feel-good story.

Nick Cafardo can be reached at cafardo@globe.com. Follow him on Twitter @nickcafardo.