Fred’s goal was ruled offside but United were awarded a penalty for a foul on Juan Mata in the build-up.
The Spaniard scored from the spot, before Romelu Lukaku doubled their lead just before half-time.
The draw for the fourth round is live on BBC One on 7 January.
It was unclear to many what referee Stuart Attwell was checking when he called on the video assistant referee in the 20th minute.
Alexis Sanchez had chipped the ball to Fred on the edge of the box to control it with his chest and pass to Mata, who then poked the ball through for the Brazilian to slot home.
The assistant referee flagged for offside against Fred before Attwell then went to VAR – and a three-minute delay followed.
Replays showed Mata was actually caught by Omar Richards before he had played the ball and eventually a penalty was given.
A message saying “VAR check in progress, checking penalty” flashed on the screen, but for those watching on TV at home and the Old Trafford crowd confusion still reigned.
“The use of VAR was questionable there – it has got to be quicker,” said former United defender Phil Neville. “In defence of Jon Moss – the fourth official – there were a lot of mitigating factors there.
“The three or four-minute delay made a sense of uncertainty at Old Trafford – even though the right decision was made in the end.”
BBC Sport’s Simon Stone added: “For the fans inside the stadium who could only see the referee and players at a distance, it was a bit frustrating judging by the whistles and jeers.
“I don’t think the time taken is an issue – if the fans know what is going on.”
United, who made nine changes from their 2-0 league win at Newcastle looked slow and disjointed.
They improved with the introduction of attacking substitutes Marcus Rashford and the lively Tahith Chong, a 19-year-old academy product who was making his first-team debut.
But they only had a couple of chances in the second half – substitute Marouane Fellaini seeing a weak effort saved and Jaakkola keeping out Andreas Pereira’s shot at the end.
United now head to Dubai for warm weather training before facing Tottenham in the league on 13 January (16:30 GMT kick-off).
Before Saturday’s game, Solskjaer said it was an “opportunity for everyone to put themselves in the frame” to face Spurs.
Based on this performance, there will be wholesale changes for when United travel to Wembley.
“The way we played wasn’t the best,” said Solskjaer. “We made nine changes, so I made it hard for the boys because they hadn’t played together too much.
“But we got the goals and managed to get through.”
- United have lost just one of their past 52 FA Cup games against teams from a lower division (won 44, drawn seven), losing to Leeds in January 2010.
- They have five victories under Solskjaer so far – their longest winning run since January 2018.
- Sanchez has been directly involved in 18 goals in 19 FA Cup games (nine goals, nine assists) in his time at Arsenal and Manchester United.
- Lukaku has scored 13 goals in his past 14 FA Cup appearances, including six in seven for United.
- Mata has been directly involved in 21 goals in 24 starts in the FA Cup (10 goals, 11 assists).