Referee Tony Bates harshly adjudged Guy Moussi to have handled a Jack Collison cross from point-blank range in first-half stoppage time, and then penalised Chris Gunter for handball after an hour.
Mark Noble scored from the spot both times and Lewis McGugan’s injury-time strike was too little, too late for Forest, who remain in the bottom three.
Allardyce said: “We went ahead through probably quite a fortunate penalty, but you have to be professional and dispose of it when you get the opportunity.
“It was not the best of days but when you can dig a win out it shows the resilience of the players.”
Southampton face Leicester City tonight needing to win to return to the top on goal difference.
Cardiff are four points behind West Ham in third following a thrilling 3-2 win at home to Portsmouth.
Kenny Miller put the Bluebirds in front, but they fell behind to goals from Marko Futacs (39) and Greg Halford (49). Mark Hudson levelled after 69 minutes and Craig Conway netted the winner in injury time.
Hull are up to fifth, behind Middlesbrough on goals scored, after beating Reading 1-0 at the Madejski Stadium as Robbie Brady netted the only goal in the 66th minute.
Boro have now gone four league games without a win after they slumped to a 3-1 defeat at bottom club Coventry.
Kevin Thomson and Julio Arca saw red for the visitors.!
Birmingham moved above Reading into the play-off zone with a 3-0 home win over Watford.
Curtis Davies (35 and 60) scored a headed double and Chris Burke made sure of the win with nine minutes to go.
Bristol City and Millwall both moved further clear of danger with victories.
The Robins were 2-1 victors over Doncaster thanks to Chris Wood (26) and Kalifa Cisse (31), while Darius Henderson scored his second hat-trick of the week – and his third of the season – as Millwall won 3-1 at Barnsley.
Blackpool moved up to seventh as they scored twice in the last five minutes to beat Crystal Palace 2-1 at Bloomfield Road. Owen Garvan’s 27th-minute penalty put the Eagles ahead, but Elliot Grandin (85) levelled and Chris Basham scored in the last minute.
Leeds climbed two places to ninth with a 3-1 home win over Ipswich. Adam Drury put the visitors in front (34), but after Alex McCarthy’s dismissal for handball the Whites stormed back through Robert Snodgrass, Ross McCormack Luciano Becchio in injury-time.
Derby’s run of four straight Championship wins ended in a goalless draw at Burnley, while a goal in each half from Will Buckley earned Brighton a 2-1 win over Peterborough.