FOXBOROUGH — Fresh off his 3-for-3 night kicking field goals in Arizona, Stephen Gostkowski had an “AFC Special Teams Player of the Week’’ certificate in his Patriots locker, mottled green-and-white paper with calligraphy-like lettering and a big gold sticker in the corner. Gostkowski did win the weekly award, but assured that the certificate wasn’t NFL-issued.

“That was a joke, that’s not real,’’ said the Patriots kicker. “Roger didn’t send that to me.’’

Gostkowski and punter Ryan Allen started to chuckle, possibly amused at the idea of commissioner Roger Goodell tinkering with fonts in Microsoft Paint.

“Roger typed that up last night,’’ Gostkowski cracked. “I’ll send it to my mom.’’

Jokes aside, the Patriots special teams unit was critical in the Week 1 win and, wouldn’t you know it, Bill Belichick’s ability to exploit the NFL rules to his advantage played a part.

Having just taken the lead, 23-21, with less than four minutes to play, Belichick elected to have Gostkowski kick the ball high instead of booting it through the end zone for a touchback. He got exactly what he wanted: Gostkowski’s kick had enough hang time for players such as Patrick Chung, Nate Ebner, and Matthew Slater to get downfield before it landed at the 3-yard line.

Arizona’s Andre Ellington was forced to make a return, getting to the 17-yard line before being tackled by Ebner. In addition, Tyvon Branch picked up a holding call on the return, setting Arizona back farther.

So the Cardinals began their final drive from the 8-yard line, the Patriots getting 17 more yards of field to work with than if they’d settled for a touchback.

“I know there was a lot of sentiment to eliminate that play, but those are the kinds of plays that are good strategy plays in football,’’ Belichick said after the game. “Certainly, we had an opportunity to kick it out of the end zone on the last kickoff, but with a good field goal kicker, a good offense, good quarterback, we try to put them on as long a field as we could.’’

The NFL has gradually tweaked rules to encourage touchbacks, moving the kickoff spot up to the 35-yard line before the start of the 2011 season and moving the touchback from the 20- to the 25-yard line this season.

The point is to minimize situations in which players are colliding into each other at full speed, with momentum built over an open field. But as the touchback provides better and better field position, at a certain point it becomes more valuable for the kicking team to roll the dice and risk a long return in hopes of pinning an opponent deep in its territory.

“We’re not going to give them a quarter or 25 percent of the field,’’ said Belichick. “We’re going to make them earn every yard that they get the ball out to.’’

Gostkowski agreed.

“We’ve got a lot of really good guys that can cover kicks that work tirelessly to be very good at it,’’ he said. “To give them opportunities to make plays, it’s kind of fun.’’

For the game, Ellington returned four of six kickoffs for a total of 61 yards. The Patriots began their four drives after kickoffs, on average, at their own 25. That was 7 yards better than the Cardinals, whose average start was their 18 yard line.

None of those Cardinals drives resulted in touchdowns, either. Arizona began its only two touchdown drives after turnovers and with excellent field position.

Gostkowski said he tries to “play dumb’’ on the sidelines and not think about situations he could get stressed out by (“He really doesn’t think that much at all,’’ jabs Allen, always by Gostkowski’s side), so when the Cardinals began their final drive, he wasn’t patting himself on the back for creating a long field.

“I just try to focus on doing my job,’’ he said, “and sometimes it works out well for you — yeah, it does. When all three phases play well in the game, that’s fun.’’

The game came down to special teams play in the most obvious of ways when Arizona’s bungled snap and missed field goal attempt let the Patriots escape with the win. The kicking game, however, had been influencing things from the start through field position.

Gostkowski said he can always empathize with a kicker who makes a mistake. The Cardinals, however, have a rookie long snapper in Kameron Canaday. Worrying about trust and communication within the kicking unit is not a problem for Gostkowski, long snapper Joe Cardona, and particularly Allen.

“I think he’s the best holder in the NFL, and to have faith and confidence in someone in front of you just helps you attack the ball a lot better,’’ Gostkowski said. “You have to play this game confidently. If you can’t, that’s when accuracy issues come into play.’’

Nora Princiotti can be reached at nora.princiotti@globe.com. Follow her on Twitter at @NoraPrinciotti.