Steve Cooper's struggling Forest side, who are now without a win in 10 consecutive league games, are third from bottom, level on 27 points with 17th-placed Everton.
Reuters reports Antony scored in the 32nd minute at the City Ground, sliding in to poke home a rebound after goalkeeper Keylor Navas's brilliant save kept out Anthony Martial's shot.
Dalot found the net in the 76th minute, scoring his first Premier League goal in his 100th appearance for United, running onto a pinpoint through ball from Antony.
"It was special. At that moment, the goal was important as well," Dalot told Sky Sports. "I think with 2-0 we were a bit more comfortable in the game. It's always special to score for this club but I'm more happy with the result."
United, who were already missing striker Marcus Rashford and defenders Raphael Varane and Lisandro Martinez, lost midfielder Marcel Sabitzer when he injured his groin in the warm-up.
Rashford and Varane are out for a few weeks, while Martinez will be sidelined the rest of the season with a foot injury.
Arsenal loosens grip on title race with draw
Meanwhile, Arsenal's Premier League title jitters returned in a potentially costly 2-2 draw at West Ham United as for the second week in a row they threw away a two-goal advantage.
The leaders were 2-0 up and cruising towards restoring their six-point cushion over Manchester City with Gabriel Jesus tapping home in the seventh minute and Martin Odegaard doubling Arsenal's lead with a volley three minutes later.
But everything changed when they conceded a sloppy penalty that Said Benrahma tucked away in the 33rd minute.
West Ham, needing points themselves to stay away from the relegation zone, were transformed although Bukayo Saka had the chance to restore Arsenal's two-goal lead early in the second half only to blaze a penalty well wide.
That miss looked even more costly moments later when Jarrod Bowen volleyed in West Ham's equaliser in the 54th minute.
Just as at Anfield last week when Mikel Arteta's team led 2-0 but ended up hanging on desperately for a 2-2 draw with Liverpool, there was a nervousness about Arsenal's play as the pressure of winning a first title since 2004 intensifies.
City's destiny is now very much in their own hands as Arsenal lead the table with 74 points from 31 games, four more than City who have played a game less and who host Mikel Arteta's Arsenal side on April 26.
West Ham, who looked the stronger side after levelling and hit the crossbar through Michail Antonio, have 31 points from 30 games, four ahead of the relegation places.
Arsenal had won all five of their London derbies away from home this season without conceding a goal and had lost only two of the previous 29 league games against West Ham.
That record looked set to continue in a sparkling start but when the fulltime whistle sounded Arsenal's players looked crestfallen at losing their grip on the title race.