Luis Robles was on hand to deny Diego Rossi on a few occasions, but the hosts did manage to open the scoring in the first-half.
Vela drove inside from the right and produced an exquisite lob over the ‘keeper on 44 minutes to put LAFC ahead.
Miami pressed for an equaliser in the second-half, but the home side looked threatening themselves and held out for their first three points of the new season.
After two years of waiting, David Beckham’s Inter Miami were playing their inaugural game in Major League Soccer, against the reigning Supporters Shield holders.
LAFC have an early sight of goal through Rossi
LAFC, who had only lost two home matches in their two seasons in MLS, started the game brightly and on the front foot.
Vela drove down the right side and crossed into the box. Rossi headed the ball back across, but Robles made a great save in the fourth minute.
Mark-Anthony Kaye had the ball in the net on nine minutes, however, it was ruled out for an offside in the build-up.
Rossi picked the pocked of the Inter Miami full-back on 28 minutes, but Robles was on hand to keep the game level with a terrific low save.
Miami’s first effort was just before the half-hour mark with Lewis Morgan’s dipping effort met by a comfortable save from Kenneth Vermeer.
LAFC drew Robles into making another top save as Rossi’s low effort took a deflection off the defender, while the visiting side went close on 37 minutes.
Hosts break deadlock after Vela lobs 'keeper
Matías Pellegrini tested Vermeer from the edge of the area. The ball was worked out to Ben Sweat, whose low cross was turned over by Eddie Segura, who needed to make the clearance with a Miami player waiting to pounce at the back post.
Inter Miami grew into the game as the first-half progressed and should’ve taken the lead on 42 minutes.
Morgan’s corner fell to the feet of Rodolfo Pizarro, who drew a low save from Vermeer. The midfielder gathered the rebound, crossed into the box from the left, and Nicolás Figal headed down and into the hands of the ‘keeper.
Two minutes later, the home side broke the deadlock after a wonderful passage of play. Vela brought down the ball and skipped inside and produced a stunning lob from 20-yards out to put his team ahead.
LAFC started the second-half on the front foot and had the ball in the net after just 33 seconds. Rodríguez played in Kaye down the left side of the box and his cross was turned home off the post by Rossi, but the goal was ruled out for offside.
Vela was the latest LAFC player to test Robles, but Miami went close to equalising when Morgan’s cross come off the back of Triston Blackmon’s head and forced the ‘keeper into making a save to avoid the own goal.
LAFC look dangerous going forward in search of second goal
Meanwhile, Morgan was involved in the action again when he swung a cross into the box, but Pizarro was unable to turn the ball home from six-yards out, striking wide of the post.
LAFC continued to look dangerous going forward, with the front three of Vela, Rossi and Rodríguez finding gaps and getting in behind the Miami defense.
The home side should’ve doubled their advantage on 69 minutes. Rodríguez played a corner into the box, but Segura couldn’t keep his shot down, blazing high over the bar.
Robbie Robinson, the first overall pick in the 2020 MLS SuperDraft, brought down a delightful pass over the top, but Jordan Harvey made a brave block to deny a certain goal.
LAFC had a chance to wrap up the points on 86 minutes, but Robles came to Miami’s rescue again as the advancing Rossi was denied by the legs of the ‘keeper.
The visitors had sights of goal through Sweat and former LAFC player Lee Nguyen, in the seventh minute of additional time, but it was the Black and Gold that held on.