Swansea, who have not won away from home in the league now in eight matches, went in at the break a goal up after Stephen Dobbie latched on to David Cotterill's long ball over the top to fire home in the 13th minute.
But Leeds fought back in the second period and just about edged a keen contest thanks to Johnson's volleyed equaliser in the 56th minute and Becchio's cool finish eight minutes later.
Simon Grayson's side are unbeaten in the league since losing at home to Derby on the opening day and have climbed to fourth place in the table.
Leeds goalkeeper Shane Higgs made his first league appearance of the season due to Kasper Schmeichel's foot injury and Max Gradel replaced Lloyd Sam in midfield.
Swansea made just one change with Andrea Orlandi stepping in for the suspended Darren Pratley.
Leeds went close to taking an early lead in the seventh minute when Luciano Becchio's shot was blocked by Garry Monk and the Argentinian headed Neil Kilkenny's corner just over the crossbar.
Kilkenny then headed over, but it was Swansea who snatched the lead via some route-one football in the 13th minute.
Midfielder avid Cotterill hoofed a long ball over the top to catch the Leeds defence flat-footed and Dobbie raced through on goal to bury his shot from a tight angle.
Leeds defender Neill Collins' header from another Kilkenny corner was cleared by Alan Tate, while Swansea midfielder Nathan Dyer forced Higgs to punch away his angled shot.
Higgs pulled off an even better save to keep out Dobbie's curling 20-yarder at full stretch just before the interval and Cotterill was also on target from 20 yards, but the stand-in goalkeeper was equal to it.
Leeds almost drew level at the end of the first period when Johnson thumped a left-footed shot from 25 yards against the crossbar.
But the 23-year-old midfielder made sure 10 minutes after the restart. Leeds skipper Richard Naylor's powerful header from Gradel's free-kick was brilliantly saved by Dorus De Vries and Johnson volleyed home the rebound, high into the roof of the net.
Grayson then sent on Sam for Sanchez Watt in a tactical switch as both sides went in search of a second.
Sam almost scored his fourth goal for the club when his stinging shot soon after forced De Vries into another smart save before Leeds edged in front with their second strike in eight minutes.
Kilkenny's rasping drive was parried at full stretch by the busy De Vries and Becchio followed up to drill home his third goal of the season.
Dobbie went close with a shot from 20 yards as Swansea responded and later brought the best out of Higgs with another fine long-range effort as the Welsh side pressed for an equaliser.
But Leeds held on for their second home league win of the season.