United host Premier League leaders Chelsea at Old Trafford on Sunday, a distant sixth from the title race.
However, Mourinho already has silverware in his possession after a Zlatan Ibrahimovic brace inspired a 3-2 EFL Cup final victory over Southampton in February and United are all square with Anderlecht after the first leg of their Europa League quarter-final.
Moyes signed a six-year contract as Alex Ferguson's successor in 2013 but failed to see out his first season at Old Trafford, with the Red Devils ending a wretched title defence in seventh.
They were beaten on penalties by Moyes' current employers Sunderland in the semi-finals of that season's League Cup – eventually won by 2013-14 Premier League champions Manchester City – and the Scot feels Mourinho is currently enjoying an easier ride for succeeding where he failed.
"You have to win trophies at Manchester United and Jose has a trophy under his belt quickly. I lost to Sunderland on penalty kicks in the semi-final," Moyes told reporters.
"It's small margins at times which can change things. That's what I’m talking about now – if we can get a small margin to go our way, it could alter things.
"But I think winning is a big thing at a club like Manchester United and you quickly learn that.
"You have to win. They will be up there competing every year. Every manager needs a bit of time and whoever is at Manchester United needs a bit of time to understand the club, the size and how it works."
Far from being a matter of small margins, Sunderland are rooted to the bottom of the Premier League – nine points from safety with six matches to play.
Despite battling back to a 2-2 draw against West Ham on Saturday, relegation to the Championship appears to be a formality.
Despite this sapping experience, Moyes – who had an unsuccessful stint in charge of Real Sociedad after his curtailed Old Trafford tenure – is confident there are better days ahead in his managerial career.
"I don't think you ever get offered the Manchester United job if you're a bad manager. You only get that position if you've had something to show previously," he added.
"Management changes. You need players around you and there are different stages in managerial careers. I think the best is still to come. I’m looking forward to winning trophies and being successful again."