Valencia, who lost 2-1 at home to Paris St Germain in the Champions League last week, were in trouble after 28 minutes when Tino Costa was sent off for swinging his arm into Tomas Pina.
The pace of Mallorca's Giovani dos Santos was a constant threat but Portugal defender Costa rose at the back post to power a header under the bar from a corner in the 60th minute.
Home keeper Diego Alves kept Mallorca at bay and the hard-working hosts were rewarded when Spain striker Soldado broke free and squeezed a shot under Dudu Aouate nine minutes from time.
Mallorca, second bottom of La Liga, have one win from their last 19 league outings.
Valencia have 40 points from 24 games, two behind fourth-placed Malaga, who beat Athletic Bilbao 1-0 at home on Saturday.
Later on Sunday, second-placed Atletico Madrid, with 50 points, visit Real Valladolid looking to close the 15-point gap on leaders Barcelona, who won 2-1 at Granada on Saturday with a Lionel Messi double.
Champions Real Madrid, in third with 46 points, host city rivals Rayo Vallecano in the late game.
In the midday kickoff, Levante stalled Real Sociedad's push for European football next season by holding them to a 1-1 draw with 10 men, both goals coming from first-half penalties.
Carlos Vela gave the Basques the lead and Michel levelled, but Levante hung on despite a second yellow card for defender Nikos Karabelas early in the second period.
Real Sociedad are sixth with 37 points.