The Russian champions failed to overcome last week's 2-0 defeat at Basel and went out 2-1 on aggregate after spurning a host of chances, including a late penalty, in the second-leg, last-16 tie on Thursday.
Italian Spalletti has been under increasing pressure from media and football pundits to resign following a streak of poor results but received a glowing endorsement from his most loyal fan, Gazprom chief Alexei Miller.
Miller, whose company has spent hundreds of millions of dollars on Zenit since taking full control of the club in 2005, sent the coach a four-line extract from a poem by famous Italian poet and Nobel Prize winner Eugenio Montale.
"Luciano, these are some words of support in a tough moment - lines from the poem 'Afloat'," Miller said on the club's website.
Miller's words meant to dispel media rumours, linking Manchester City manager Roberto Mancini and Dynamo Moscow coach and former Chelsea full back Dan Petrescu with the Zenit job.
Spalletti, who last year extended his contract with Zenit until June 2015, was confident of turning things around.
The team also lost 1-0 to Rubin Kazan last weekend when the Russian season resumed after a three-month break, leaving Zenit eight points behind league leaders CSKA Moscow with 10 games left.
"I see this match as a turning point," Spalletti, who has led Zenit to back-to-back Russian league titles in the last two years, told reporters after the game against Basel.
"Despite what has been said and written about our team we showed strong character, a fight to win and I believe that in the coming matches we'll play even better than we did today."
However, Spalletti's passionate plea did not stop some experts from calling on Zenit management to sack the Italian.
"Spalletti, it's time for you to go back to Roma," former player Valery Reingold said in reference to the Serie A club, which the Italian had coached for four years before joining Zenit in 2009.
Ex-Zenit and Russia striker Alexander Panov had a similar view: "I think Spalletti has exhausted himself at Zenit. You can see it by just looking at him - it seems he just doesn't have the fire anymore."
Spalletti's job of rebuilding Zenit will not be easy as the team is still trying to overcome a visible discontent among several senior players unhappy after the club splashed out more than $100 million on Brazil striker Hulk and Belgium midfielder Axel Witsel just before the transfer deadline last September.
"It's obvious, Zenit's internal problems have not subsided, and this just goes on and on and on," Reingold said.
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