Rashford limped off in the 80th minute during United's 2-0 Premier League win over Everton at Old Trafford, and will miss the first leg of their Europa League quarter-final clash against Sevilla.
United have not provided a timeframe on their top goalscorer's return.
"It's a setback and he's disappointed, but he's not totally broke because he will return quickly. He's started his rehab and that helps to get back soon," Ten Hag told reporters ahead of Thursday's home game.
"(Rashford will miss) A few games... We'll have to see it how it develops."
Rashford's injury will be a big blow to United, and the club will be hoping Anthony Martial (27) can step up in the England international's absence.
Ten Hag backed Martial and dismissed criticism of the French forward, who has grabbed seven goals in 17 appearances this season and has spent much of the campaign on the sidelines injured.
Martial made three substitute appearances for United in their last three league games, scoring against Everton.
"I think he is ready to start..., the front line did very well against Brentford and we picked the same against Everton," Ten Hag added.
"The advantage was we could bring Martial back slowly, giving him minutes, but I think already against Everton he was ready to start the game.
"The time he needs for a goal is less. When he's in the team, we play our best football and we have our best results as a team."
United are also missing the services of winger Alejandro Garnacho (18) and defender Luke Shaw (27), but the manager expects the England left back to be "back on short notice".
The Old Trafford club are fighting for silverware on multiple fronts this season, and remain alive in the Europa League and FA Cup.
They have also won the League Cup and are fourth in the league standings on 56 points from 29 games.
"We are playing really good football. We've already won one cup and we're all enjoying it, the dressing room is close together and the players are giving their all in training and games," goalkeeper David de Gea (32) said.
"We are still in three competitions and we're in the right way."
