Less than a week after they pulled back a 10-point deficit to secure a last-gasp draw with Bath on home soil, Leicester scored three second-half tries in Italy having trailed 13-8 at the break.
Treviso then led 19-13 with half an hour remaining at the Stadio di Monigo but the Tigers hit back to set up a huge Heineken Cup clash with Ulster at Welford Road a week on Sunday.
Tait said: "We got what we set out to achieve from the game and that was maximum points.
"At half-time we were in a bit of a hole but it's a mark of this team that we've been in that position a few times this year and we've managed to dig ourselves out of it.
"That's testament to the group. We've got some really experienced guys in there and players who put us in the right parts of the field, certainly in the second half.
"If we had been on our game from the start we wouldn't have been in that position, though.
"By our own standards, the first half wasn't really good enough.
"We knew that Treviso are a very physical team and they would make it difficult but we need to have a look at ourselves because our standards will have to be a lot higher against Ulster.
"The last 30 minutes were a better reflection of how we wanted to play."
Leicester now welcome pool leaders Ulster to Welford Road knowing that a victory of any sorts will see them finish top of Pool Five and secure a spot in the quarter-finals of a tournament they last won 12 years ago.
Tait added: "That was a massive win. If we didn't win here, it would have made next week largely irrelevant.
"Taking maximum points sets us up for next week and that's how we want it. It's in our own hands.
"Ulster are playing very well at the moment but it's a winner-takes-all game at Welford Road and it should be a cracking game.
"We know what's riding on it and we know how tough it's going to be. But that's why you come to the big clubs - you want to play in great competitions and great matches."