With this victory, Vettel extended his title lead over Lewis Hamilton to 25 points, after Hamilton completed the race in seventh postion.
Australian Daniel Ricciardo completed the podium by finishing in third, after starting the race in fifth position.
The race began with Ferrari's locking out the front-row of the grid, with Raikkonen in pole position - and after an incident-free start to the race, the Finnish driver took control of the race, with his teammate Vettel close behind.
Due to the narrowness of the track, opportunities to overtake were limited on the Monaco circuit - so teams had to utilise pit stops to their advantage in order to move up the pack.
Ferrari chose to call Raikkonen into the pits from first, which allowed Vettel to get some clear air out the front of the pack - while Raikkonen re-joined the race in traffic, slowing him down significantly.
This allowed Vettel to use the 'overcut' on his team-mate and emerge from his pit stop in the race lead, much to the frustration of his Finnish counterpart.
Daniel Ricciardo also utilised a similar strategy to move up two spots, passing the Mercedes of Valtteri Bottas and his Red Bull team-mate Max Verstappen with a late overcut.
After each driver had registered their pit stop for the race, Vettel took command - and pulled his lead out to as much as 12 seconds, before a safety car on Lap 60 brought the pack together when Pascal Wehrlein's Sauber was pressed against the track barriers.
However, when racing resumed seven laps later - Vettel flew back out into the lead and eventually crossed the chequered flag to claim his third race victory of the season.
Raikkonen crossed the flag 3.1 seconds off the pace, while Daniel Ricciardo was only 0.6s away from second place.
Mercedes' Valtteri Bottas was the next to finish in fourth, while Red Bull's Max Verstappen ended up in fifth.
Carlos Sainz (Toro Rosso) finished in sixth, narrowly ahead of Lewis Hamilton - who rose seven spots to finish in seventh.
Romain Grosjean (Haas), Felipe Massa (Williams) and Kevin Magnussen (Haas) rounded out the top ten.