A Michail Antonio goal sent Nottingham Forest into the Capital One Cup second round at Prenton Park as Rovers played their first game under new owners Mark Palios and his wife Nicola.
Edwards conceded there were chances to draw level after Forest defender Michael Mancienne's second-half red card but was left to lament his side's lack of cutting edge.
"I have met the new owners and as a couple they want to take the club forward as quickly as they can in a realistic way," said Edwards after his side's 1-0 defeat.
"It is a really good football club and it should be playing at a higher level.
"If we work together and get the backing the supporters have given us over the last two games then I think we can move the club on.
"There is a little bit more money in the transfer kitty but it isn't ideal timing with regards to recruitment.
"We do have a thin squad so if we can add more depth then I think we have the quality to do something this season.
"In the second half the game obviously changed with the sending off and we created a number of half chances without ever making a real clear cut opportunity.
"We did manage to hit the bar but it wouldn't quite go in for us."
Tranmere's best chance to draw level came when Danny Woodards rattled the visitors' crossbar in the second half but Forest eventually closed out the game confidently.
And their assistant manager Steve Wigley was pleased to see the Championship side progress after what he believed to be a contentious sending off.
"We probably deserved to get another goal but we didn't and that produced a few scary moments, particularly when we went down to 10 men," said Wigley.
"Obviously we have to change things a bit at that point and Tranmere tested us but the lads showed real character and we got the job done.
"We wanted to win to carry on the momentum from Saturday but I was very disappointed to see Michael Mancienne sent off. I thought he was very unfortunate to get the first booking but he was always going to get booked for the second challenge.
"We rested a couple of players and gave others a chance to get some game time. We have had some key players leave the medical room and get back on the training ground in the last couple of weeks and that has made a big difference.
"One by one I expect them to be trying to force their way into the side."