Blackburn, making his first start since May 16 after spending more than two weeks on the DL with a strained left quad, allowed two runs on five hits over five innings to pick up his second win in eight starts.
"It was exactly what I needed," said Blackburn, who entered the game with an 8.47 ERA. "I needed to go out there and get a good one under my belt. I think confidence is the main thing, if I can go out there and do something like that. Obviously, I'm not trusting my stuff right now but tonight was a big gain to trust my stuff more."
Blackburn's main problem was his pitch count, as he needed 92 pitches to get 15 outs. But Twins manager Ron Gardenhire trusted him to face Mike Moustakas with a runner on and two outs in the fifth, and Blackburn was able to get him to fly out to right to end the inning.
"We need to see that, we need him to step up," Gardenhire said. "He's supposed to be one of our big starters, and we need him to start eating up innings and winning ballgames for us. That'll help us get on the right track ourselves."
Blackburn was able to overcome a rocky start, allowing a leadoff single to Alex Gordon in the first before quickly serving up a two-run homer to Yuniesky Betancourt.
But Blackburn settled down and the Twins were able to take advantage of the Royals' bullpen, as right-hander Felipe Paulino recorded just two outs before exiting the game with a strained right groin muscle.
Minnesota scored in the first on an RBI double by Ryan Doumit off reliever Luis Mendoza, after Paulino left the game due to the injury.
"It was just not my day today, unbelievable," Paulino said. "But the good thing right now is it's no big deal. My groin is just a little bit strained and hopefully tomorrow there will be less inflammation in that spot."
The Twins re-took the lead in the fifth, when Morneau hit a two-run single off Mendoza with two outs and the bases loaded.
"He stayed on the ball, I think he hit it off the end of the bat a little bit, but he got it to the grass," Gardenhire said. "That's all you have to do in that situation."
The Twins added an insurance run in the seventh on an RBI single by Jamey Carroll after Ben Revere tripled against reliever Kelvin Herrera.
The bullpen did the rest, as Jeff Manship, Jared Burton, Glen Perkins and Matt Capps combined to throw four scoreless frames.
Burton was especially impressive, as he struck out three of the four batters he faced. He ended up striking out eight of the nine batters he faced on the road trip.
"My put-away pitches are just really locked in right now," Burton said. "I'm executing them. I'm getting ahead and then going into put-away mode, and it's just working right now. Early in the year it was mainly my changeup; I'd say the last couple weeks my slider's starting to feel a little better, so I have another option to go to."
The victory gave the Twins their third straight series win, as they swept the A's and took two out of three in Cleveland before taking two out of three in Kansas City. It marked the first time this season that Minnesota won three series in a row.
"If we get our rotation kind of rolling and in a groove, it could be fun," said Capps, who improved to 14-for-15 in save opportunities. "We dug ourselves a pretty big hole, we know that. There's nothing we can do about it other than now enjoy the off-day and get ready for Friday's game."