Unai Emery’s side, who finished a distant third behind Barcelona and Real Madrid last season, were ahead in the first minute when Soldado slotted home the rebound following a Pablo Hernandez strike.
Racing were level six minutes later when the Valencia captain nodded the ball into his own goal from a corner and Lautaro Acosta then finished off a fine breakaway to make it 2-1 to the visitors in the 14th minute.
Adrian gave Racing a two-goal cushion 11 minutes after the break before Valencia’s French defender Adil Rami pulled a goal back a minute later.
Soldado made amends for his earlier blunder by scoring twice in two minutes, including a last-gasp winner, to snatch a rousing victory.
La Liga is getting underway a week late after last weekend’s first round of matches fell victim to a strike called by the players’ union over unpaid wages, a dispute with the league that was finally resolved on Thursday.
Those games, including Barca at Malaga and Real at home to Athletic Bilbao, will now be played in January.
Champions Barca begin their bid for a club record-equalling fourth straight title at home to Villarreal on Monday (1900 GMT) while Real Madrid visit Real Zaragoza on Sunday (1800).
Racing’s defeat marked a losing return to the Mestalla for Argentine coach Hector Cuper, who led Valencia to successive Champions League finals in 2000 and 2001 when they lost to Real Madrid and Bayern Munich respectively.
In the opening game of the campaign earlier on Saturday, Real Sociedad held on with 10 men to secure a 2-1 win at Sporting Gijon who also had a player sent off late on.
Granada, back in the top flight for the first time since 1976, host fellow promoted side Real Betis in the late kickoff.