Lobjanidze stunned a star-studded Miami line-up with two long-range strikes either side of half-time to fire Atlanta into a 2-0 lead.
Argentina superstar Messi threatened to inspire a comeback for the home side after pulling a goal back to make it 2-1 but Atlanta restored their two-goal cushion when Jamal Thiare poked home Caleb Wiley's cross in the 73rd minute.
While Lobjanidze grabbed the glory with his brace of goals, Messi's World Cup-winning Argentina teammate Thiago Almada was the star of the show for the visitors, setting up two goals with some superb individual skill and tormenting Inter's suspect defence all night.
Despite the defeat, Miami remain top of the Eastern Conference with 34 points from 17 games, after nearest rivals Cincinnati also lost on Wednesday, falling 2-0 at home to Nashville.
Miami, who started Messi along with former Barcelona teammates Luis Suarez, Sergio Busquets and Jordi Alba, were always struggling to gain a foothold against a vibrant Atlanta line-up who had gone into the game without a league win since March 31st.
Atlanta gave an early warning of their attacking intent when midfielder Brooks Lennon flashed a low shot just wide after only 10 minutes.
Lobjanidze handed Atlanta the lead just before half-time with a superb individual goal, surging forward from midfield before unleashing a shot full of pace and precision into the top corner past Miami goalkeeper Drake Callender.
Lobjanidze, who is due to be part of Georgia's squad at next month's European Championship, bagged his second on 59 minutes after brilliant trickery from Almada launched an Atlanta counter-attack. Lobjanidze's curling shot took a slight deflection before beating Callender.
The second Atlanta goal stung Miami into life, and it was Messi who gave the Eastern Conference leaders a lifeline with a goal out of nothing on 62 minutes, darting forward and steering a low, long-range shot into the bottom corner.
But Atlanta continued to threaten on the counter-attack and made it 3-1 on 73 minutes when more good play from Almada allowed Wiley to cross for Senegalese striker Thiare, who prodded into the net.
