The visitors won the one-day international series 3-2, but their final victory was overshadowed by the dismissal of Buttler, who was run out at the non-striker's end by Sachithra Senanayake, who took the bails off when he strayed out of his crease.
Neither player is involved in the five-day series but as captain Mathews made the call to uphold the appeal, a decision that incensed the England side, it has added extra spice to the series.
The incident also came after Sri Lankan heckles were already raised by the reporting of Senanayake over a suspect bowling action.
Mathews and his opposite Alastair Cook spoke briefly after the match about the incident, when feelings were still high, but the 27-year-old said his side was more interested in winning the series than provoking tensions.
"We want to be focused, concentrate on what we can do," Mathews said .
"There are so many things happening around but we want to stick to our basics, try and do our best and win against England.
"We played pretty well in the ODIs and confidence is pretty high. We all know it is going to be a hostile series, but we are up for the challenge and we just want to enjoy ourselves and play some good cricket."
Talk inevitably returned to the controversial run out, with Mathews refusing to back down on his belief that Senanayake was in the right.
"I don't want to see that happening again. Buttler was warned a few times - he did it in the Lord's ODI and we analysed it after that, and they have taken 21 twos in the last 10 overs," he said.
"He was backing away but we did nothing about it - we just couldn't let him do that in every game. We warned him twice and, as I keep saying, I don't know how to stop a batsman doing that consistently so we had to go for it.
"I don't want to keep talking about it - we've been talking about it ever since the ODI series so we just want to put it to bed and move on."
In Mahela Jayawardene and Kumar Sangakkara, Sri Lanka have two of international cricket's most experienced batsmen, albeit ones who have struggled on previous tours to England, but their seam attack, of vital importance early in the northern summer, is less so.
Of the five pace bowlers named in the 15-man squad, Chanaka Welegedara is the only one to have taken a Test wicket in England, and together with Shaminda Eranga, Nuwan Kulasekara, Nuwan Pradeep and Dhammika Prasad they have played a combined 69 Tests.
England are expected to name three debutants, with Sydney-born Sam Robson to open with Cook, all-rounder Moeen Ali to bat at number six and Chris Jordan one of the pace options.
It is coach Peter Moore's first Test back in charge of the England since replacing Andy Flower after the 5-0 Ashes humiliation in Australia.
With Kevin Pietersen's international career over, England have bigger concerns than a controversial run-out as they look to rebuild for the future.
"It's an exciting time," Cook said. "Three new caps is a very exciting time and we have a lot of guys with not a lot of experience but then on the other side of the changing room you've got five or six guys who have played close to 100 Tests.
"It's quite a big split but the last couple of days (training) have been brilliant."
He backed the newcomers to show that they could be the future of the England side.
"It takes time to feel settled in international cricket, you never feel totally comfortable but you need a bit of experience to know how it works, to feel the difference and the step up from county cricket.
"That can lead to a bit of inconsistency at the start, look at all the senior guys, they've had their ups and downs to get to where they are now.
"There are also guys who have had amazing starts to their careers, hundreds on debuts, five-fors, you're going into the unknown and the opposition don't know that much about you."
Cook said Gary Ballance will bat at three, followed by Ian Bell, Joe Root and Moeen, who will be also expected to chip in at the bowling crease with his part-time off-spin.
The second and final Test begins at Edgbaston on June 20.
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- Alastair Cook
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