Moyes, without the services of former Liege player Marouane Fellaini, dropped Joseph Yobo and drafted in Baines and chose Louis Saha as Yakubu's partner in attack as he made two changes to the team that lost to Liverpool at the weekend.
Despite reports of player dissatisfaction over their manager's delay in signing a new contract, the visitors began the game full of energy and commitment, snapping at the heels of the hosts.
Saha, making only his second start since signing from Manchester United, and Mikel Arteta were both exhibiting their full range of tricks in the opening exchanges, but the defenders undid all their hard work upfront midway through the first half.
Keeper Tim Howard did well to keep out Steven Defour's dipping shot, but he was left stranded when Jovanovic cleverly floated the ball back into the six yard box.
After a great run at the back post Belgium's Witsel was left unmarked and he blasted the ball into the net with a low volley to send the home supporters wild.
Everton now needed to score twice, but as the rain lashed down they gave themselves hope thanks to good work by Victor Anichebe.
Just a few minutes after coming on as substitute for Tony Hibbert, the Nigerian's shot was deflected over with the equaliser arriving from the following corner.
Liege keeper Rorys Aragon dropped the ball twice and Jagielka was lurking a few yards out to stab home.
Although Everton were making life difficult for the hosts, Liege were always a threat on the break and it was a dazzling run by Jovanovic that ended with the referee pointing to the spot.
Jovanovic carved open the Everton defence with a one-two with Dieumerci Mbokani, before falling under a tackle from Baines.
It was a tight decision with the Everton defender appearing to get the ball before making contact with the attacker, but Jovanovic took full advantage to calmly convert.
Everton's players poured forward in search of an equaliser, but not even the presence of keeper Howard in the opposing box could force the game into extra-time.
Moyes, his players and fans looked stunned at the final whistle and they will now have to pick themselves up for the weekend's game against Newcastle.
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