"I think Forest were the better team over 90 minutes," Coleman admitted.
"We looked jaded and tired. Our passing was not the best and it wasn't really until they scored that we kicked into life."
Mark Crossley, returning to the club where he spent 13 years, was the busier of the two goalkeepers in a lacklustre first-half, twice denying Andy Reid.
However, the 35-year-old was unable to prevent Forest from taking the lead when King fired home after pouncing on a mistake by Carlos Bocanegra in the 71st minute.
Andy Cole then miss-kicked in front of goal and Steed Malbranque dragged his shot wide as Fulham desperately chased an equaliser.
But fears of an embarrassing exit were overcome when Radzinski, who was making his first start for a month after recovering from a hamstring injury, slipped Luis Boa Morte's pass under Paul Gerrard in the 85th minute.
Fulham might even have won the game in normal time had Boa Morte showed greater conviction when he sprinted clear with seconds remaining, but his hesitation proved costly.
"I won't tell you what some of the lads said to him after the game," said Coleman.
However, it took Fulham only three minutes of extra-time to take the lead. Radzinski cut in from the right before nonchalantly placing a left-foot shot into the top corner.
Substitute Brian McBride then added a third, and although Reid clawed a goal back for Forest, Cole lifted the ball over Gerrard to see Fulham through.
