The home side, trailing by 203 on first innings, resumed on 85-9 and last pair Jeetan Patel and Olly Hannon-Dalby took the score to 115 before the latter missed a forward push at Adil Rashid and was bowled.
Patel, who had just lifted Rashid straight for six, was left unbeaten on 49, having hit five fours and one six, while the continually-impressive young seamer Ben Coad ended with 5-50 to earn himself match figures of 10-102.
Yorkshire’s resounding victory was an emphatic response to their surprise defeat to Hampshire in their opening game last week.
“It was a big week for us having played so well last week when we didn’t get anything out of the game,” said Gale.
“I thought we showed a lot of character.
“After last week’s game we spoke a lot about winning key passages of play. In the third innings we let Hampshire back in the game, but in this game we got an opportunity to get ourselves in front and get away from them and we took it.
“If I was being ultra-critical I would have liked one of our top six to go on and get a hundred after they all got starts.
“But, on the whole, winning those key sessions was the difference.”
Yorkshire’s next Championship match is against Hampshire at the Ageas Bowl on Friday when they will have prized England duo Jonny Bairstow and Joe Root available.
While there is good news there, captain Gary Ballance will continue to have to make do without three key bowlers, with seamers Jack Brooks, Liam Plunkett and Ryan Sidebottom still injured on the sidelines.
There had been hope that Plunkett and Sidebottom might be available for the trip to the south coast, but the rematch with Hampshire has come just too soon.
Brooks, on the other hand, was always going to be out injured for a longer period of time.
“Ryan and Liam are another week away,” said Gale.
“Ryan was going to play in the second team this week, but it’s come about that he’s still not quite right.
“Liam is still feeling it a bit, so we can’t risk him.
“Fingers crossed, he will be available for the 50-over stuff. I’m pretty sure he will be available for the first game of that (away to Nottinghamshire on April 29).”
After travelling to Hampshire, Yorkshire will then turn their attention to their final Championship game before the Royal London One-day Cup starts in earnest at Trent Bridge.
They then resume their four-day campaign against Lancashire at Emirates Old Trafford on May 19.
“The break will give Ryan a good block of training, so when he does come back he’ll be ready to go,” added Gale.
“I don’t think we can afford to rush a guy who is nearly 40 years old. If he breaks down, it could be the end of his career. We have to do what’s right by him.”
Warwickshire, meanwhile, must dust themselves down from successive innings defeats from which they have taken just three points.
It has been a bracing start to the regime of sport director Ashley Giles and first-team coach Jim Troughton and one that has hurt a set of experienced players, which was viewed in some quarters as potential title-challengers.
The first two games have supplied a serious reality check, ahead of another tough fixture, at home to Surrey starting on Friday.
“The first two games have been really disappointing after such high hopes during the winter,” said Giles. “We’ve done a lot of good preparation, but just not put it into practice. It’s certainly hit the dressing room hard, but all we can do is try to turn it around.”
Elsewhere, defending champions Middlesex started the defence of their Division One title with a hard-fought draw against Hampshire.
Both sides took 10 points from the Ageas Bowl, but the 2016 champions were staring down the barrel of a nail-biting conclusion when a three-wicket new-ball strike left them 215-7.
Had Hampshire rolled over the tail an achievable run chase would have been left, but tail-enders Toby Roland-Jones (25), Ollie Rayner (17no) and Tim Murtagh (19no) dug deep.
In the end the sides shook hands at 278-9 declared, Kyle Abbott’s good form continuing as he claimed 5-59.