Dave Attwood and Matt Carraro were the players who eventually touched down but Davis was full of praise for Biggs' contribution after a lame 22-6 defeat at Gloucester a week previously.
"He did have a big game - probably his his best of the season," said Davis. "If he gets the ball in open play, he's a very dangerous runner.
"We had a chat together during the week about how we could get him more involved. If he does get the ball at speed he's a hard man to stop."
Bath sneaked victory with a Tom Heathcote penalty late in the game - but would have lost out if Leicester tryscorer Jeremy Staunton had added to his 25-point haul with a last minute penalty.
Davis continued: "The response we got after the Gloucester defeat in training and in this match was outstanding.
"Full credit to the guys. They obviously didn't want to go through another week like the one we've just been through.
"The performance and the end result is the direct outcome of all the work we've put in over the last week."
He added: "To be fair to Leicester, they came out of the blocks really quickly. What we had to do was just stay in the game and ask a few questions.
"We made two or three line breaks but just couldn't beat their last defender. A little bit of frustration crept in and we had one or two forced errors but as the first half wore on we started to get a bit stronger.
"Any score on the stroke of half-time always makes your team talk that much easier. The guys stuck at it and it was a typical Bath-Leicester game."
While clearly disappointed at coming away from Bath with only a losing bonus point, Leicester director of rugby Richard Cockerill praised his players for their efforts.
"Our attitude was outstanding," he said. "We had 20 people missing before Scott Hamilton pulled out late on through illness. We'll see who's available for next week.
"We battled hard and did enough to win, I thought. But you can't keep giving the ball away and kicking poorly out of your own half.
"I've got no real criticism for our lads. There are lots of positives but we don't want to be sitting here with just a pat on the back."
Cockerill hopes to have some of his World Cup players back in action next week after the quarter-finals had been decided.
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