Moyes has had to put up with a degree of criticism since he took over as United manager after he oversaw a poor start to the season.
The Red Devils have turned a corner of late, but they still sit in eighth position, eight points behind leaders Arsenal.
Similar expectations fell on Villas-Boas' shoulders when he took charge at Chelsea following his success at Porto, but his reign lasted 257 days after a schism emerged between the young manager and some senior players at the west London club.
Villas-Boas believes handling such pressure is part and parcel of being a manager of a top club.
When asked whether there had been too much "drama" over United's start to the season, the Tottenham manager said: "Drama enough? I don't think so.
"For the champions of England to sit in their position I don't think there has been a lot of drama, to be fair.
"I was involved in another club before and there was much more drama surrounding my results than Man United's results."
United lost to Manchester City, Liverpool and West Brom early in the campaign, but they have won their last three games in all competitions.
Moyes insists he can lead United up the table, and Villas-Boas is sure the club will be snapping at Tottenham's heels in the push for a top-four place soon.
"Manchester United have played most of their difficult fixtures, so things are looking a bit more positive. They'll get back eventually," Villas-Boas said.
"They look to be coming back strong and achieving the results they need.
"I would suppose they would join, eventually, the rest of the teams.
"The distance is nothing for the Premier League so at the end of the season I would say they will be there competing for that first place."
Villas-Boas visits Moyes' old club Everton on Sunday hoping to build on the momentum gained in back- to-back wins over Hull.
Tottenham were in the midst of a strong run last season when they visited Goodison Park and all looked to be going well when the Londoners took the lead halfway through the second half.
Everton hit back with two late goals to steal the victory, however.
Those three points would have been enough to edge Tottenham in to the Champions League last term, so Villas-Boas knows his team cannot afford to throw points away again this season.
"We've spoken about (last year's) result as the deciding factor for us not making it into the Champions League because everything changed quite dramatically," Villas-Boas said.
"It serves us as a lesson to be kept aware of and it's not something we want to see again."
Villas-Boas' side will have to stop striker Romelu Lukaku if they are to come away with something from Goodison Park this time around.
Lukaku has scored five goals in seven games since joining on loan from Chelsea, where he used to work under Villas-Boas.
The Belgian was furious with his treatment by Villas-Boas during his time at Chelsea, but the two have since buried the hatchet.
Villas-Boas has been impressed with how he has developed since his successful loan spell at West Brom last term.
"The player did extremely well last season and has continued to do well for Everton. The potential is there," Villas-Boas added.
"We all recognise he can become an influential striker in the future.
"Chelsea's standards are very, very high so the jump will be very difficult for him, but he has shown he can cope with the Premier League and it's another step up for him, at Everton."
Villas-Boas will be without Kyle Naughton and Nacer Chadli after they suffered injuries in the Capital One Cup win over Hull on Wednesday.
Andros Townsend has recovered from his nasty fall against the Tigers last weekend but will wear a protective splint on his arm.