Swan was chosen by the Mets in the third round of the 1972 draft. He spent most of to in the minor leagues with only brief (and poor) stints in the majors. One of his better early performances came on May 11, 1974 when he personally out-played the opposition by collecting a career-high three hits, knocking in one run, and scoring another at the plate while allowing no runs on four hits over six innings on the mound. Later that season, he broke his elbow and missed significant playing time. He returned successfully and, in , was named the International League Most Valuable Pitcher.
In , Swan's numbers declined slightly but so did the rest of the team's as number one starter, Jerry Koosman, lost 20 games and the Mets sank into last place in the division.
In 1978, the Mets struggled to a 66-96 record but Swan started the season with a five-hit complete game shutout, the third of his career. In the second game of a July 4, 1978 doubleheader, Swan logged a career-high 13 strikeouts but surrendered two ninth-inning runs to lose the game 3-2. The loss dropped his record to 1-5 despite a very good 2.66 ERA. Swan's ERA continued improving and he won his next seven decisions. On September 16, he allowed one run and three hits over nine innings but was again denied a win. At season's end, Swan had the best ERA in the National League as well as the second-best WHIP and second-best hits per nine innings. His ERA was only 1.67 at Shea Stadium. Despite his excellent statistics, he finished with only a 9-6 record, the Mets were the third-worst pitching team in the league, and they finished with the worst record in the N.L. was another good season for Swan and was also his most active. He set career highs in innings pitched, games started, complete games, shutouts and BFP. Swan was the lone bright spot for the Mets' pitching staff. His 14 wins were not only a career-best but were more than any two of his teammates combined. The Mets finished with 99 losses, again last in the National League.
After his two good seasons for the Mets, Swan signed a contract which made him the highest paid pitcher in Mets history at the time. In response, he began with an even better ERA than either 1978 or 1979. In mid-June, he was 5-4 with a 2.21 ERA for another terrible Mets team. After losing his next four decisions, Swan was placed on the disabled list in mid-July with what turned out to be a torn rotator cuff. He made two more starts a month later but pitched poorly in the first start, left the second start after only two innings, and was done for the season. He started with a loss but left his second game after only one batter in a freak play. Swan fractured his rib when Ron Hodges -- the Mets' catcher at the time -- nailed him in the back with a throw when the second batter of the game attempted (successfully) to steal second base. As a result, Swan landed on the disabled list. He returned more than a month later and made two relief appearances before the 1981 Major League Baseball strike cancelled the next two months. When play resumed, Swan was back on the disabled list and made only one more start the rest of the season.
After missing almost all of 1981 with a rotator cuff injury, and the aforementioned broken rib and a baseball strike, Swan returned well in and finished second to Joe Morgan for the N.L. Comeback Player of the Year Award. His 11 wins were again the highest total on the team and he was the only pitcher on the team with a winning record and more than three decisions. For the first time in his career, he accumulated significant time as a relief pitcher and managed a fantastic 1.30 ERA in the new role. Swan even hit the only home run of his career on August 4 (although the event was drowned out when his teammate, Joel Youngblood, made history the same day by becoming the only player in major league history to play for two teams in two cities in one day).