But Everton - who Moyes led for 11 years before leaving in 2013 - did enough to get the points that keeps Sunderland in serious trouble courtesy of Idrissa Gueye's opener five minutes before half-time and Romelu Lukaku's record equalling second ten minutes before the end.
What a day for battling Bradley Lowery as he was roared on to the pitch before Sunderland's match at Everton.
"Yeah, [we want] the highest position but I like to play for something". We know the team behind us is West Brom, who we play at home in two weeks, and above there is still a gap.
Finishing above Liverpool in the Premier League table is Everton midfielder Ross Barkley's objective.
"As a striker you want to score as much as you can but the games are the most important thing", he said.
"It was the rebuilding of a football club and we went on to qualify for the Champions League with a very limited budget".
It was a missed chance which Sunderland would regret as, from the resulting corner, Everton broke away and Lukaku held off Billy Jones to score and secure all three points.
More misery for Sunderland as Everton boost European hopes
But I also think he can improve: giving options to the midfield players and threat to the defenders. A few teams have had theirs. "That experience might help us".
Lukaku's goal was his 60th Premier League strike as he equalled Duncan Ferguson's club record for top tier goals which has stood since 1992. They will have their chances, and I think they might find a way of picking up a point.
Barkley made headlines by celebrating prior to slotting home his most recent goal against Bournemouth, but the midfielder insists the gesture was not meant to be disrespectful.
Sunderland's Jermain Defoe crashed an effort against the crossbar with 15 minutes left, shortly before Lukaku held off Bryan Oviedo to bundle home a second.
"But they do have someone like that and that's the only way they are going to get out of this situation".
"I was not trying to take the mickey out of Bournemouth, it was just more the enjoyment of the game".
"I have incredible memories of Everton but I also go back knowing I had a great chairman and great people around me", said Moyes of his former club.