Jones, the No. 15 overall pick in the 2021 NFL Draft, completed 29 of 39 passes for 281 yards and a score in the 17-16 setback Sunday at Gillette Stadium in Foxborough, Mass.
His 74.4% completion percentage was the highest in history for a rookie who threw at least 25 passes in his first start.
Jones said he needs to learn how to protect himself better, be more vocal with teammates and release the ball quicker.
"I just tried to stick to what I was taught, and we can do a better job, I can do a better job just going through my reads and communicating," Jones told reporters.
"I'll improve on my communication, and we'll make sure that we're better. It wasn't good enough."
Jones averaged 7.2 yards per attempt, with a long completion of 26 yards. His 102.6 quarterback rating ranks 12th among the 29 quarterbacks who threw at least 20 passes in Week 1 games played through Sunday.
"He definitely has a lot of poise," Patriots running back James White said of Jones. "He stayed composed throughout the game. Obviously, everything wasn't perfect, but he kept us in it, keeping everybody motivated in the huddle.
"It wasn't our best football game on offense, definitely got to play better in the red zone and create more points to help the defense out."
Jones threw his first pass attempt backward for a fumble, which the Patriots recovered, but improved as the game progressed.
The Dolphins took a 7-0 lead on a three-yard touchdown run from quarterback Tua Tagovailoa with 5:35 left in the first quarter. Jones started a 14-play, 75-yard drive at the end of the quarter. Patriots kicker Nick Folk ended that possession with a 27-yard field goal.
Jones gave the Patriots a 10-7 lead two drives later when he threw a seven-yard touchdown pass to wide receiver Nelson Agholor.
Dolphins kicker Jason Sanders made a 48-yard field goal on the next drive to tie the score at halftime. Tagovailoa's Dolphins opened the second half with a nine-play, 75-yard touchdown drive. The second-year quarterback connected with rookie wide receiver Jaylen Waddle for a three-yard score to end that possession.
Jones led a 14-play, 77-yard drive on the next possession, but the Patriots ended the drive with another Folk field goal.
Defensive back Jonathan Jones intercepted Tagovailoa to the end of the Dolphins next drive, but the Patriots could not capitalize on the turnover. Running back Damien Harris fumbled at the Miami 11-yard line and the Dolphins recovered the loose ball to secure the victory.
"We've just all got to do a better job," Patriots coach Bill Belichick said. "Mac competed hard. I thought we had a lot of guys competing hard. We've just got to perform better as a team."
Coach Brian Flores and and Dolphins defenders were impressed by Jones' performance.
"He didn't really make any mistakes," Dolphins defensive back Jason McCourty said. "He was poised. Even watching him in the huddle getting guys in, yelling, commanding.
"A lot of what he looked like in the preseason is how he came out and performed. He gave his team a chance to win."
Jones told reporters that he plans to improve his leadership when he returns to training this week.
"I think being the quarterback, I need to do a better job, demanding better in practice," Jones said. "Sometimes I just let things slide, myself included. That's not good enough.
"So I'm going to try and be more vocal, and I am a vocal person, but it's kind of a new situation, and I can be better. But it starts with me. ... We've got to do better on offense. So we'll do that."
The Dolphins tied for allowing the second-fewest passing touchdowns and tied for the league-lead in interceptions in 2020.
The Patriots face the New York Jets in Week 2. Carolina Panthers quarterback Sam Darnold completed 24 of 35 passes for 279 yards and a score, and rushed for a touchdown, against the Jets on Sunday in Charlotte, N.C.
The Jets allowed the second-most passing yards in the AFC and third-most passing scores in the NFL last season.
"I think [Jones] is a guy that loves the game of football and I think he is a really resilient player," Agholor told reporters. "He stays in the game and plays hard and is only going to get better."