Neither Morecambe nor Newport were able to find the back of the net in normal time, but in the second half of extra time the deadlock was broken by Carlos Mendes Gomes' penalty as the Shrimps preserved their perfect Wembley record to make club history.
Adams took over at Morecambe with the club second from bottom of the English Football League in November 2019 but has led them to new heights and their first taste of League One football.
However the Scot, who has been linked to other clubs including Bradford in recent months, would not be drawn on his own future at the helm.
"There's no deal done with any club. There's a lot of speculation on players' futures and managers' future. All I can say is that there's no deal done," Adams said.
"I think today is all about making history. We've done that exceptionally well throughout the season. I've got a contract with Morecambe and that's what we are playing to."
The Spaniard sent goalkeeper Tom King the wrong way after referee Bobby Madley had pointed to the spot when substitute John O'Sullivan was brought down by Ryan Haynes, but the contact appeared to be outside the area.
In the absence of VAR, however, the decision stood.
Adams said: "I thought it was a penalty kick. Our player gets across their defender and he's lucky not to get sent off or booked.
"It was a clear penalty kick. You only have to look over the season. They've played against us with 10 twice and won twice.
"But we always knew 11 v 11 that we'd have a very good opportunity. Sometimes you get what you deserve and today that happened."
Newport had a penalty shout themselves turned down in the first half, when Shrimps goalkeeper Kyle Letheren appeared to make contact with Scot Bennett's head, but the referee that time waived play on.
Newport manager Michael Flynn took aim at the decisions, saying the absence of VAR for the crucial clash was a "cop out".
"I said it two years ago, they've got to have VAR at these big games. They'll come back with the excuse 'we haven't had it all season so it's unethical'," the County boss said.
"That's a load of rubbish. There's not one game in the season, other than the play-off final that determines which league you're going to be playing in. So that's a cop out.
"It's two terrible decisions and I'm gutted for my players, for the club and the supporters.
"Their goalkeeper's come out missed the ball completely and punched our player in the face, our number 17 Scot Bennett, and nothing's given.
"Then the referee has seen a foul, a so-called foul, which even if it was a foul was outside the box and he's given the penalty.
"It's two terrible decisions and I'm not going to say too much because I'm going to get fined. It's two terrible decisions and we need VAR for these big games, something's got to be done."