Johnson's second goal of the game and fourth this season came just one minute after Dallas completed a comeback from a 2-0 deficit when Michel drilled a corner kick from the right side into the upper left roof of the Seattle net to tie it.
The Sounders (4-3-3) stretched their unbeaten streak to six games (4-0-2) after a franchise-worst 0-3-1 start.
"For the first couple games of the season, my body was beat up, and it was frustrating, because there was a time that as a team, we faced a really difficult challenge," Johnson said. "But it was just getting that rhythm and getting that confidence, and staying true to myself and believing in my teammates."
League-leading Dallas (7-2-3) hadn't lost since a 3-1 setback at Chivas USA on March 10 in the second game of the season.
"You never want to squander a two-goal lead, but we knew they were dangerous on set pieces," Sounders coach Sigi Schmid said. "The timeliness of our third goal was vital, and we continued to get opportunities."
Johnson put the Sounders on the board in the 16th and Lamar Neagle made it 2-0 in the 30th. Second-half substitute Obafemi Martins added an insurance goal in the 83rd minute as Seattle racked up four goals for the second straight game.
Johnson's winner was set up by Lamar Neagle, who sent a through ball toward the top of the penalty area. Johnson took it around onrushing goalkeeper Raul Fernandez and, from 10 yards in front of the net on the right side, sent a shot across the goalmouth and into the far side of the net.
"We had opportunities in the first half, but we just had to be more composed, show more composure in the box when we got our chances, and stick the ball in the back of the net," Johnson said. "We got the chance in the second half, and Oba and myself, we made it count."
Kenny Cooper got one back for Dallas early in the 54th before Michel's equalizer in the 61st.
"Being able to measure ourselves against the first-place team in our conference is important," Schmid said. "Being able to show that once all of our weapons are healthy we can score goals is important, as well."
Midfielder Osvaldo Alsono set up Johnson's first goal on a counterattack. Johnson took the long pass from Alonso, went into the penalty area and fired into the back center of the net from 16 yards in front, getting his shot past onrushing defender Matt Hedges.
That was just the second time in 12 games this season that Dallas has allowed a first-half goal.
Neagle scored on a wide-open breakaway 14 minutes later after taking a through-ball from Brad Evans.
FC Dallas cut it to 2-1 when Cooper, playing near the penalty spot, headed in a free kick that Michel drove into the box from 40 yards out.
Michel's dramatic game-tying goal came on a corner from the right side that went just over the outstretched fingers of Sounders keeper Michael Gspurning.
"They created a lot of chances, a lot of breakaways, a lot of counterattacks," Dallas coach Schellas Hyndman said. "It seemed like the only time Seattle really put their foot on the ball is when we were down a man. The rest of the time, it was very fast-paced. I think probably the bigger thing that mattered was I think our guys lost their composure and emotions when they considered some of the calls that weren't being made, and then, you're just disappointed."