German Mukhtar opened the scoring in the 11th minute, drilling home a low shot from inside the box, but four minutes before the interval Nicholas Gioacchini levelled for the visitors, bundling home after a corner.
The game turned decisively in the 68th minute when Kyle Hiebart brought down Jacob Sheffelburg in the box and was sent off with a second yellow card.
Mukhtar converted the penalty to restore Nashville's lead and then wrapped up the win with a first-time shot from the edge of the box.
The 28-year-old Mukhtar is now top scorer in MLS with 13 goals in 18 appearances and with the win, Nashville close the gap on Eastern Conference leaders FC Cincinnati to five points, having played one game more.
The third match without a win leaves Western Conference leaders St. Louis now level on points with reigning champions LAFC.
The Californians had Mexican veteran Vela to thank as they snapped a three-match losing streak with a controversial victory at Kansas City.
An Alan Pulido penalty had given Sporting the advantage at the break but three minutes after the interval, Honduran Denil Maldonado, who had given away the spot-kick, made amends with an angled header.
In the final minute, Vela was found with a ball over the top and showed all his experience with a calm finish into the bottom corner.
Kansas City thought they had levelled in stoppage time through a header from substitute Roger Espinoza but to the fury of the home team the goal was ruled out after a VAR review spotted a foul in the build-up.
After last week's wonder goal from inside his own half from the Crew's Lucas Zelarayan, Real Salt Lake's Pablo Ruiz provided a carbon copy with an audacious goal from over 50 yards out at D.C. United.
Wayne Rooney's side had been down to 10 men after Steven Birnbaum was dismissed in the 26th minute and things got worse when Danny Musovski made it 2-0 before Chris Durkin pulled one back for D.C.
The 2021 champions, New York City, remain outside the playoff positions in 11th place after they needed a late equalizer to earn a point at home to the Columbus Crew.
Christian Ramirez had put the Crew in front in the 48th minute from close range but in the fourth minute of stoppage time Gabriel Segal grabbed the leveller.
The New England Revolution beat Orlando City 3-1 in a restricted program in MLS due to the international break.
The United States face Canada in the final of the CONCACAF Nations League on Sunday.
