England left-back Bridge had put pen to paper on a four-and-a-half year contract at Eastlands earlier in the day following a big-money move from Chelsea.
City were already missing Robinho and Stephen Ireland from the start, but their afternoon went from bad to worse when they also lost Shaun Wright-Phillips in the 17th minute.
Early injuries to two players shaped the outcome of this contest. While Forest coped well with the loss of the their captain James Perch after just 17 minutes following a tackle from Vincent Kompany, City found themselves desperately short of ideas after Wright Phillips limped off.
Already without the attacking prowess of Ireland and Robinho, City were punished for their slack defending after 38 minutes when Micah Richards' weak clearing header from Wes Morgan's cross fell to Nathan Tyson, who was arriving at the edge of the area.
Tyson's stunning first-time left-footed volley beat Joe Hart at the far post and put City firmly on the back foot.
Forest capitalised on the lacklustre City defence four minutes later when Chris Cohen won the ball off Pablo Zabaleta and found Matt Thornhill in the area with a cross.
Thornhill's mis-hit shot was going wide, but Robert Earnshaw was on hand to tap his effort past Hart and in off the post.
Things improved slightly after the break with Felipe Caicedo testing Paul Smith in the Forest goal twice shortly after the restart and laying the ball off for Daniel Sturridge to have a go, but there was little joy for City as the game became more fractured and drifted away from Mark Hughes' side.
The 75th minute was the signal for home fans to leave after substitute Joe Garner scored Forest's third goal.
In inexplicable fashion, a Forest corner bypassed everyone in the area to go out for a throw-in on the opposite side.
Dietmar Hamann took the resulting throw-in, but Garner couldn't believe his luck when the intended throw to a City defender fell short at the feet of the Forest player, who needed no further invitation to turn and beat Hart and put the tie out of reach.
Boos rang out from the remaining home fans at full-time as City were left to contemplate the real possibility of missing out on European football next season.