Chris Brown's last-gasp goal cancelled out Mathias Pogba's first-half strike to take 10-man Crewe to penalties.
But Robbie Blake smacked the crossbar with his penalty as the Alex, who had Matt Tootle sent off in injury-time for two yellow card offences, held their nerve and tucked away every spot-kick to win 5-3.
A virus had wiped out the Rovers squad, forcing boss Saunders to make five changes to the team that lost to Oldham in Saturday's FA Cup clash.
But the Welshman was delighted after his team took Crewe to the wire.
"We've had a virus running through the camp," he said. "We've had something like 16 missing or ill and we've only got 18, so work that out.
"I tried to get the game called off (on Monday) because we couldn't field a team but rules are rules and I've had to put players out there who weren't fit so that we didn't get fined as a club.
"I couldn't name the team until I got to the ground and we haven't trained or been through anything so I didn't expect us to win. But the players were magnificent and we showed desire to come back and played some great football.
"I'm sure some of the fans have gone home pretty excited about some of the young players we've got."
