Former Barcelona and Real player Saviola joined Malaga on transfer deadline day two weeks ago and on his first start for the side burst through on to Isco's pass to score in the 27th minute.
Saviola, known as 'El Conejo' (the rabbit), then laid on Joaquin to score the second after Levante had grabbed an equaliser against the run of play when Michel ran the length of the pitch to score just after halftime.
Malaga lost defender Sergio Sanchez to a red card near the end but Francisco Portillo made sure of the points with a volley in time added on.
Malaga set out on their first Champions League venture when Zenit St Petersburg visit on Tuesday, while Levante debut in the Europa League at home to Helsingborg two days later.
Valencia's new coach Mauricio Pellegrino can breathe a little easier after new recruit Aly Cissokho headed the winner from a free-kick at the Mestalla for their first three-point haul of the campaign.
Sofiane Feghouli had given the hosts a fourth-minute lead as he capitalised on some sloppy defending, but Celta quickly pegged Valencia back.
They deservedly levelled when Gustavo Cabral bundled a volley past Valencia keeper Diego Alves.
Valencia improved in the second half after France defender Cissokho struck in the 51st minute, but they will need to play much better if they are to take something from their Champions League opener at Bayern Munich on Wednesday.
Barcelona won 4-1 at Getafe while Real Madrid lost 1-0 at Sevilla.