This is a complement for the great solution from
LifeExpectancy and what to do if the achievement doesn't unlock even if you did everything right.
FIRSTLY, SOME IMPORTANT TIPS:*** Spoiler - click to reveal ***
1) You must play at least on Heroic difficulty solo or co-op. In case of co-op, the achievement should unlock for all players but this is buggy and may not unlock for some players. I'll explain how to solve this later.
2) You can start from Rally Point Bravo, but you'll need to do the first part of the mission. The time will start counting just after the second part of the mission, after the cutscene. You can skip the cutscene.
3) You can die many times and still get the achievement. When you die, the mission time keeps counting but the time for the achievement resets from the last checkpoint, anyway, it is very important to do everything ASAP.
THE ACHIEVEMENT:*** Spoiler - click to reveal ***
When the second part starts, rush for the engines and destroy all 4 quickly, after this, in your left, you'll see about 7 Banshees spawning. Kill them all and hide yourself close to your frigate, leaving it to do the most part of the job while you take out the lone Banshees and Seraphs. (This useful tip was given by Crocsx as a comment on IatrogenicFGC solution)
Make sure to use the red waypoint markers around the edges of your HUD to locate the last Banshees and Seraphs and destroy them quickly. Then, rush for the corvette’s deck and wait the order from radio to land on it.
If you done everything fast and right within 3 minutes, the achievement must unlock after you destroy the last enemy or after you regain the control on corvette's deck, after the cutscene. Unfortunately sometimes this achievement takes a long time to unlock, or simply doesn't unlock at all.
IF THE ACHIEVEMENT DIDN'T UNLOCK FOR YOU, DO THE FOLLOWING (CONSOLE ONLY):1) Pause the game and choose "
Save/Quit". Keep pressing

until the main menu of Halo: TMCC appears and finally close the game from the dashboard:
Press

, select the game, press

and choose "
QUIT". You'll return to main screen of console dashboard.
2) With game totally closed, delete your local save (just the local save, NEVER CHOOSE "
Delete everywhere"):
Select the game, press

and select the option "
Manage game & add-ons". Go to "
Saved data", on the right side, choose your save (it will have your gamerpicture and gamertag). With your save selected, press

and choose the option "
Delete from console", just the local save will be deleted.
3) Go back to main screen of console dashboard pressing

. From here, you must became offline:
Press

, go to "
Settings" » "
General" » "
Network settings" and choose "
Go offline".
4) Play offline again until you complete the achievement requirements exactly as it was explained above in "
THE ACHIEVEMENT:".
5) After the cutscene, you land on the corvette’s deck and you regain the control, pause the game, choose "
Save/Quit", go to Halo: TMCC main menu and quit the game from the dashboard.
6) Repeat the step "
2)", deleting your local save again.
7) Now, repeat the step "
3)", but this time you must select "
Go online".
If you done everything right, the achievement must unlock when you are online, still on the dashboard. If this doesn't happen, just start the game normally, wait until the game loads the main menu. Press

on main menu and wait a couple minutes. During the wait process, you can play any mission solo or any kind of multiplayer game, just keep the game running while you wait.
My friend
rafa1000house helped me with this solution of deleting the local save. I did everything this way and I unlocked the achievement on my first try. Good luck!