Edwards was the most successful West Indies bowler with five wickets for 104 runs from 26.5 overs in the Australia first innings total of 431, and he was thrust into the role of spearhead after Taylor was sidelined with a stiff shoulder just prior to the start of the Test at Sabina Park.
"Jerome was a big miss and also Chris (Gayle)," Edwards told reporters Friday night.
"Going out there to bowl, I thought, not only for myself, but for the team, I really had to put in a big effort to bowl Australia out."
This was Edwards' sixth haul of five wickets or more in 32 Tests and he admitted it was easily his best performance against the world's top side.
"Australia have a lot of good batters and you have to put in an extra effort to get the guys out," he said.
"I thought I was getting most of my balls in the right areas. This is what the coach has been asking of us. The pitch was playing up and down, but this is the best I have bowled against Australia so far."
"I just had to be positive and do what I know how to bowl and stick to the team plan."
Edwards feels that he is now bowling more consistently than in the past, when his slingy action did not always allow him to bowl accurately.
"The skipper wants you to do a job and you find yourself going out there and giving a bigger effort being the main strike bowler in the team," he said.
The only blemish on Edwards' performance was that umpire Russell Tiffin warned him for running on the protected area of the pitch.
"It made me bowl a lot wider and a couple of balls were off target," he said.
"It is just for me to go back in the nets and work on this. I tried to focus and work on my rhythm during the last series I played as, when my rhythm is off, I tend to bowl the ball all over the place."
Edwards believes the second day belonged to West Indies, after they captured the last six Australia wickets for 130, but the home team finished on 115 for three - after Stuart Clark worked his way through their top order with three wickets, including the prized scalp of West Indies captain Ramnaresh Sarwan.