United moved up to sixth in the table with a 3-0 win over Everton at Goodison Park, as Alejandro Garnacho scored a remarkable overhead kick before Marcus Rashford and Anthony Martial put the match to bed after the break.
It means Ten Hag's side are just four points off the top four, despite the United boss being under pressure after a stumbling start to the season, and only six points off the pace set by Arsenal at the top.
United this season have a better record than any other side in matches against teams in the bottom half, but they have lost all four of their games against clubs currently in the top ten of the Premier League. They will get the chance to address that in their new two League games, as they travel to Newcastle next weekend before hosting Chelsea a few days later.
Ten Hag was full of praise for his players after their win over Everton, hailing Bruno Fernandes' leadership qualities after the United captain, the club's usual penalty taker, handed over the ball for Rashford to take the spot-kick and end his long goal drought.
However, Keane was less than impressed with those comments, and criticised the upbeat mood from the United camp as he demanded the team start to deliver against their rivals for European places.
"He's giving toffee out there to Bruno for passing on the ball for the penalty," Keane told Sky Sports.
"Absolute bloody rubbish. Absolute rubbish coming out with that. Manchester United are sixth now - if you went back a few years, if you were sixth you'd be embarrassed. But obviously they seem happy with sixth.
"They have a long way to go. Manchester United have to be competing with the best teams. That's including Liverpool, City, Arsenal, and obviously you've got teams like Spurs having a decent spell, Villa and of course Newcastle.
"They go to Newcastle next week - we'll see what Manchester United turns up. That's what they're supposed to do, turn up for the big games."