Wolves goalkeeper Aaron McCarey, a former Walsall loanee, was the hero - saving from Craig Westcarr and Andy Taylor - as Wolves won a penalty shoot-out 4-2 at Molineux following a 2-2 draw after 90 minutes.
The hosts had twice taken the lad through Liam McAlinden and Sako only to be pegged back by Wanderers old boy Ashley Hemmings and substitute Troy Hewitt.
Sako had been the target of bids from Nottingham Forest before the closure of the summer transfer window and did not play at Port Vale last weekend as he was not in the right frame of mind due to the speculation over his future.
But he looked fully committed against Walsall, and Jackett said: "He assured me over the weekend he would be 100 per cent with me, the club and the players and he did prove that with a very good performance.
"Sako was 100 per cent with it for this game and knew where he was and wanted to play. It didn't take any persuasion from me.
"I spoke to him and speaking to him was a positive conversation and he was very enthusiastic to play.
"He put a lot of effort in and I was pleased with that as the dynamics of the squad are that you need matchwinners.
"We are a work in progress and we want to add and not take away."
Wolves have now won their last five games in all competitions, and Jackett added: "You could say that we did it the hard way but there is no easy way in professional football.
"There was a lot of plusses from the game. At times I thought we played some fantastic football and we could have been more goals up."
Meanwhile, Jackett has been shortlisted for League One's manager of the month award for August alongside Peterborough boss Darren Ferguson, Leyton Orient manager Russell Slade and Coventry's Steven Pressley. The winner will be announced on Friday.