The Andalusian side, UEFA Cup winners in 2006 and 2007, failed to qualify for continental competition last term for the first time in almost a decade, dealing the cash-strapped club a significant financial blow.
They started this season poorly but since the sacking of coach Michel in January replacement Unai Emery has revived their fortunes.
Monday's success lifted them to 10th on 35 points with 12 matches left, six below a Europa League berth and eight behind Malaga who occupy the fourth Champions League spot.
"It's my first hat-trick in the league, the Cup and every other competition," Negredo said.
"We have to be strong here at home and away too and it was important to win and keep climbing up the table," added the 27-year-old, who was given a rousing ovation by the home fans when he was substituted with just over five minutes left.
"This is the dynamic that we have to keep up now, we played very well tonight."
Negredo fired Sevilla ahead in the 12th minute at a rain-soaked Sanchez Pizjuan stadium when he was perfectly placed to slot home after Celta goalkeeper Javi Varas had palmed away a Jesus Navas cross-shot.
The pacy Navas was a constant threat down the right and another centre from the Spain international in the 29th minute ricocheted out to Negredo just inside the area and he blasted the ball into the net.
Celta, who are 18th on 23 points, fought back strongly towards the end of the first half and a defensive error gifted them a goal five minutes before the break, Augusto Fernandez firing home after smart work from Iago Aspas.
Varas should have coped better with a swerving effort from Sevilla midfielder Gary Medel in the 66th minute but could only palm the ball into the net before Negredo pounced on another loose ball to complete his hat-trick two minutes later.
Sevilla have now won four and drawn one of their seven league games since Emery took over, with the two defeats coming at leaders Barcelona and champions Real Madrid.
Celta Vigo had 46.7% of the possession during the match. Celta Vigo had 14 shots, including four on target.
The four attempts by Iago Aspas were the most of any Celta Vigo player.
Roberto Lago was brought on to replace Carlos Bellvis in the 81st minute when the latter went down with an injury.
The combination of Javi Varas to Mario Bermejo was the most common during the match, as the two players connected 15 times.