In what was a big opportunity to build momentum, Town found themselves 2-0 down to the out-of-form visitors, who had not won in 11 league games, inside just 15 minutes.
Ricketts' side rallied after half-time and Aaron Pierre and Callum Lang struck within 16 second-half minutes as the hosts restored parity and looked in the ascendancy.
But Mellon's men, who are fighting for survival after just three points from 11 games, kept going and were rewarded through Corey Blackett-Taylor's winner five minutes into added time.
Ricketts, whose side failed to build on Saturday's win over Doncaster, said: "I'm really, really disappointed, annoyed, angry at being 2-0 down. Two attacks and we're 2-0 down.
"You can't do that, you can't give a team something to hang on to.
"We made a few changes at half-time and within a quarter of an hour the team were back in it and that's credit to them.
"For the next half hour or so we had some unbelievable chances to get that third goal and go ahead but couldn't take them whether that's a block or missing the target when we really should score."
Town missed golden chances to go 3-2 up with Josh Laurent guilty of missing the target with away goalkeeper Scott Davies busy.
But Mellon's visitors were keeping Salop honest in an open encounter and the electric pace and cool finish from Blackett-Taylor means it is two steps back for Ricketts' side.
Supporters reacted frustratingly as Shrews fell short in all departments against the side who had not won in League One since December 21.
"No-one's more frustrated and disappointed than me at losing a game," Ricketts added.
"The players are the same. We desperately wanted to win again tonight to back up Saturday's performance and we didn't.
"We started 2-0 down and then we got into a rhythm and the performance was good.
"We're pushing and really want to take the next step and win the game, it was there for us, but I'm disappointed.
"I'd be more disappointed if we weren't having the shots and clear opportunities to win."