Rashford has been a player transformed under Ten Hag and carried his fine form for England at the World Cup into the Premier League's return.
"Our players performed really well in the World Cup, I was happy to see that and now they continue," Ten Hag told Amazon Prime. "They have to stay on that level, keep going and progress from game to game."
A brilliantly worked corner opened the floodgates for United as Christian Eriksen picked out an unmarked Rashford inside the area to sweep into the top corner.
Forest were without their first choice goalkeeper Dean Henderson as he is on loan from United and his understudy Wayne Hennessey was culpable when the home side doubled their lead just three minutes later.
Rashford's burst down the left and teed up Martial at the edge of the box, but the Frenchman's strike should not have had enough power to beat the Welsh international goalkeeper.
United were also short-handed at the back with Lisandro Martinez absent following Argentina's World Cup celebrations, Diogo Dalot injured and Harry Maguire still feeling the aftereffects of an illness.
As a result Raphael Varane was forced into starting just nine days after playing 113 minutes for France in the World Cup final, while Luke Shaw lined up as a centre-back in a back four for the first time.
But Forest did little to test a makeshift backline as they remain with just one goal away from home in the league all season.
Steve Cooper's men did have the ball in the net shortly before half-time, but Willy Boly's effort was ruled out for offside after a lengthy VAR check.
Gakpo no go
The home side should have added more than one goal in the second half, leading Ten Hag to repeat his call for a new striker.
United appear to have already missed out on one target with Cody Gakpo set to join rivals Liverpool from PSV Eindhoven.
"We are looking for a striker because games are coming every three days," added Ten Hag. "It's tough and we want to play in all the leagues. Hopefully we can keep going till the end of the season, so we need numbers in the squad."
Antony fired straight at Hennessey after fine play from Rashford and Bruno Fernandes gave the Brazilian a clear sight of goal.
Hennessey also stood firm to block from Martial and Rashford from close range, but United finally got the third goal they deserved three minutes from time.
The imperious Casemiro strode forward to win possession and picked out his Brazilian compatriot Fred to slot into the far corner.
Victory takes United to within one point of the top four and with a game in hand to come on fourth-placed Tottenham.
Forest remain rooted in the relegation zone, but just one point adrift of safety.