"It is 2-0 at half-time," said Liverpool boss Klopp. "You have to be completely on alert, 100 percent in the right moment and play the second half like we played the first. We know it will be a tricky atmosphere for us."
In stark contrast to the free-flowing thrills of Manchester City's 4-3 win over Real Madrid in the other semi-final, on Tuesday, Emery's men followed the same tactic that saw them past Juventus and Bayern Munich by retreating behind the ball.
However, the Yellow Submarine could barely lay a glove on Klopp's men as Liverpool just had to bide their time to make their dominance count.
"The 2-0 is a clear, deserved result," said Emery. "It could have been even worse but we still have a chance.
"I want to show we are capable of playing a different game at home."
The Reds have now scored 135 goals this season, but were held scoreless in a frustrating first 45 minutes.
Mane headed wide Mohamed Salah's pinpoint cross with the best opening.
The Senegalese also saw a deflected effort fly just wide and Villarreal goalkeeper Geronimo Rulli escaped unpunished with some sloppy handling from a number of speculative efforts.
Another long-range strike was the closest Liverpool came to breaking the deadlock before the break as Thiago Alcantara's sweet strike clipped the post.
The six-time European champions started the second half at a higher tempo and quickly began to punch holes in Villarreal's rearguard.
Fabinho's goal was ruled out for offside against Virgil van Dijk in the build-up.
Moments later, Anfield erupted as the error-prone Rulli finally cost the Spaniards with a feeble attempt at parrying Henderson's deflected cross.
The last thing the visitors could afford was the floodgates opening, but they were powerless to stop the waves of Liverpool attacks.
Salah was the architect with a perfectly weighted pass for Mane to prod past Rulli on 55 minutes.
The offside flag came to Villarreal's rescue once more to rule out Andy Robertson's strike.
Emery was hailed the "king of the cups" by Klopp for his stunning Europa League record, with one of his four triumphs in that competition coming as Sevilla boss against the German during his first season at Liverpool in 2016.
However, Liverpool are a different beast six years on and the limitations of a side sitting seventh in La Liga were shown up as they struggled to mount any sort of response to going behind.
Luis Diaz smashed across the face of goal as Liverpool threatened to kill the tie off with a third.
The English giants had to settle for a two-goal advantage, but they are yet to lose by a two-goal margin in 55 matches this season.