Torrential rain made conditions difficult at the Estadio de San Mames, but Raul Garcia headed in the only goal in the first half.
Athletic Bilbao had the better chances, with Aritz Aduriz close to adding a second with a shot that was just wide.
Rodrigo came closest for Valencia, but saw his effort blocked.
"Both teams played with fantastic spirit in difficult conditions," said Neville. "They were probably the worst conditions I have seen a game of football played in.
"It was very difficult for our forwards to get into the game, but we kept fighting."
Elsewhere, holders Sevilla played out a goalless draw with Basel in Switzerland.The Spanish side had midfielder Steven Nzonzi sent off late on.
Fellow Spanish side Villarreal took command of their tie with Bayer Leverkusen as Cedric Bakambu's brace gives them a 2-0 lead heading into the second leg in Germany next Thursday.
Shakhtar Donetsk are also in a strong position to reach the quarter-finals after beating Belgian side Anderlecht 3-1 in Lviv.
Taison and Olexandr Kucher scored twice in the first half for the Ukrainian side and although Frank Acheampong pulled one back for Anderlecht, Eduardo's late finish put Shakhtar in control.
Meanwhile, Marco Parolo gave Lazio a potentially crucial away goal as the Italian side drew 1-1 at Sparta Prague, while Mehmet Topal struck in the closing stages as Fenerbahce beat Portugal's Braga 1-0 in Istanbul.
Gary Neville's Valencia suffered a narrow defeat by La Liga rivals Athletic Bilbao in the first leg of their Europa League last-16 tie.