Carlos Vela set the visitors on their way and rounded off the scoring late in the second half, while Imanol Agirretxe netted before half-time to double the visitors' lead and Asier Illarramendi grabbed his first goal since returning to the club.
The Basques move up to 14th place in the table with this morale-boosting win - three points clear of Levante, who slip down to 19th.
Moyes was without striker Jonathas de Jesus and defender Diego Reyes for the trip to Valencia as both players were sent off in last week's dramatic 2-0 defeat to Atletico Madrid, but his side got off to a strong start.
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Vela squandered an opportunity to open the scoring within five minutes, but did give his side the lead just moments later.
The Mexican was played in by a wonderful through ball by Agirretxe and beat Levante goalkeeper Ruben Martinez at the second attempt.
The veteran Basque striker then got a goal of his own, appearing unmarked in the area to nod home a cross from Xabi Prieto.
The hosts improved somewhat in the second half but could not beat goalkeeper Geronimo Rulli, and were dealt a killer blow by Illarramendi with nine minutes remaining.
The midfielder got a final touch to Chory Castro's shot to send it beyond the reach of Ruben and just three minutes later the Uruguayan was the provider again, setting up Vela's second.
Levante's misery was then compounded in the last minute as striker Deyverson was shown a second yellow card for diving and dismissed.
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