The former Sky Blues defender took charge on a temporary basis after Andy Thorn was sacked on Sunday, and got off to the perfect start as he auditions for a job he admits he would like.
Coventry were behind inside four minutes to a Peter Lovenkrands header, but two goals in three minutes from Cody McDonald and Kevin Kilbane gave the hosts the lead.
Blues levelled before the break through Jonathan Spector and that was how things stayed until the end of 90 minutes.
But Carl Baker's strike seven minutes into the added period sent Coventry into the next round at the Ricoh Arena on Tuesday night.
"A home draw against one of the big boys would be lovely. It would bring a big crowd and financial rewards, which are much needed," Shaw said.
"I was in this position before at Millwall but I wasn't ready then, I was still playing and still enjoying playing.
"I would be lying if I said I didn't want the job. I would like it but that is up to the powers that be and I want the focus to be on the players.
"All I can do is try and get the players prepared in the right way and let them know about the opposition, if I do that then I have done my job. If the players can take that out there with them then they have half a chance.
"I told the players the first half against Bury was very good, we played good football with passing sequences but tonight we did the ugly things as good as we have done all season."
Morgaro Gomis hit the bar three minutes after Baker's goal but that is as close as Lee Clark's men came to drawing level again.
"It doesn't matter whether we deserved it or not we lost it and we will continue to lose if we give goals away like that. We won't win many games if we need to score four every time. It is outrageous," Clark said.
"It is down to poor decisions by experienced players. Believe it or not, it might not look like it but we do a lot of work on the defensive side in training.
"We have got to respond better to setbacks, we are too helter-skelter at the moment.
"I am missing the two best full-backs in the league, when I get them back I will give a team to be proud of.
"We've had a horrendous 10 days. Never mind shooting ourselves in the foot we are blowing our leg off at the moment, it's soul destroying.
"This is the biggest challenge of my managerial career but I'm not going to shirk the challenge."