Kansas City, top seeds in Major League Soccer's Western Conference, saw Johnny Russell, Ilie Sanchez and Khiry Shelton convert in the shootout to snag the win.
Melia, meanwhile, dived left to deny Oswaldo Alanis then went right to stop both Jackson Yueill and Cristian Espinoza.
Melia improved to 6-0 in career shootouts. Four of those came with Sporting and two with lower-division Rochester and Charleston against MLS competition in the US Open Cup.
The 34 year-old went into the contest having allowed just 14 of 26 penalty attempts he has faced, the lowest percentage in MLS history.
"Everybody out here can hit penalties, but the one thing is, Timmy can save them," Sporting coach Peter Vermes said. "I've never seen that before, in any game, ever."
Melia said penalty shootouts are "different than penalties in a game."
"You're trying to pick up little cues, you're trying to see what directions people are looking, tendencies of the shooter, their run-up, their approach," he said.
"I think it's just another piece. You try to be the most complete goalkeeper you are and you try to work on all different facets of your game."
Kansas City, who missed the post-season last year as they finished 11th in the West, booked a Western Conference semi-final showdown with Minnesota United, who defeated the Colorado Rapids 3-0.
Neither Sporting nor the Earthquakes could find a goal in extra time after a frantic finish to regulation.
Sporting appeared to have won when Gianluca Busio scored one minute into second-half stoppage time.
But Chris Wondolowski, MLS's all-time leading goalscorer, made it 3-3 seven minutes into injury time, drifting to the back post to head home a left-footed cross from Cristian Espinoza.
Although Wondolowski has a league-record 166 regular-season goals, it was just his second in the post-season.
Busio had put Sporting up 3-2, the 18-year-old seizing upon a ball flicked back from Shelton and firing a right-footed shot into the goal.
Sporting had taken the lead in the fourth minute with a Roger Espinoza's header off a corner kick.
Sanchez also scored for Kansas City on a set-piece in the 47th minute, making it 2-2.
San Jose had responded to Sporting's early goal with two first-half strikes.
Cristian Espinoza crossed from the right flank to set up Carlos Fierro, who fired a left-footed shot that spun in off another player in the 22nd minute.
San Jose took a 2-1 lead in the 34th, Cristian Espinoza again playing a role as he raced up the right side and found Shea Salinas near the penalty spot.
- Molino fuels Minnesota -
At Allianz Field in Saint Paul, Minnesota, Kevin Molino scored twice and Robin Lod added another as the Loons notched their first-ever MLS Cup playoff victory.
Trinidad and Tobago international Molino opened the scoring in the 22nd minute with a left-footed shot past Rapids keeper William Yarbrough.
Finland's Lod, put through by Emanuel Reynoso, made it 2-0 in the 54th when he curled a left-footed shot into the corner.
Molino put it out of reach in the 79th, latching onto a pass from Jan Gregus and chipping over the onrushing Yarbrough.
In the late game, FC Dallas eliminated the Portland Timbers in a shootout after the teams played to a 1-1 draw in regulation.
Standout matches today