The England man, enjoying a purple patch following the World Cup, netted his 18th goal of the season in the sixth minute to silence an expectant crowd at the City Ground.
New United loan signing Wout Weghorst scored his first goal for the club just before halftime, to put the visitors in total control, and Bruno Fernandes added gloss to the scoreline late on.
Erik ten Hag's men are hot favourites to reach Wembley as they seek to end their six-year trophy drought, with either Newcastle or Southampton lying in wait.
"I'm happy with the performance and I think over 90 minutes we controlled it," Ten Hag told Sky Sports.
But he highlighted a disallowed Forest goal in the first half, warning his side could not afford to let opponents back into the match when they were in charge.
The Dutchman was careful to avoid any hint of complacency ahead of the second leg at Old Trafford on February 1.
"We have to play one game and we have to do the same. We have to prepare, we have to make a good game plan and the players have to be focused," he said.
Former Ajax boss Ten Hag has revitalised United in his first season in charge but will be desperate to win silverware to appease the hungry fans.
Brazil midfielder Casemiro was back in the United starting line-up after being suspended for Sunday's 3-2 Premier League defeat at leaders Arsenal, which left United fourth in the table.
Forest, with new boy Danilo in the starting line-up, came into the match full of confidence after just one defeat in their past seven Premier League matches had lifted them to 13th in the table.