This answer was originally written for Firefox Legacy. At least one new add-on for Firefox Quantum has been found and suggested at below.
Use this Firefox Add-on called Download Manager Tweak (DMT). This seems to be the only add-on that works with recent Firefox releases (after Firefox 30).
Follow the steps to configure DMT in Firefox Legacy:
- Go to Add-ons Manager > Extensions (or press Ctrl+Shift+A)
- Look for Download Manager Tweak and click on its Preferences, then a dialog window "Download Manager Tweak Options" will appear
- In File/Window - Download Manager window, change from
Open in a separate window to Open in a tab
- In Advanced - Advanced Options, deselect/remove option for
Load the Download Manager tab in the background
- Click Close to finish
The default setting runs DMT download manager in a separate window. By following above steps, when user initiates a download, the DMT download manager will open in tab and switches immediately to the tab (This behavior can be changed in step 4).
DMT download manager: The add-on has its own download manager that is different from the built-in. The location bar will show chrome:// instead of about:Downloads for Firefox current built-in ones.
Works with Firefox 56 and earlier (Deprecated), but no longer available from Firefox Add-ons. Tested with Firefox 41.0 on Linux. No screenshot.
For Firefox Quantum (version 57+)
Download Manager Tweak (DMT) has since not been rewritten yet for Firefox Quantum. As of today (late 2018), at least one new add-on offers similar feature like DMT.
Download Manager (S3) by Oleksandr
View and manage downloads from a tidy status bar
This is a new life in Firefox 26+ for old well-known addon Download Statusbar
New Features: Replace Downloads Library page.
Now you have full control of your downloads history!

Follow the steps to configure S3 in Firefox Quantum:
- Go to Add-ons Manager > Extensions (or press Ctrl+Shift+A)
- Look for Download Manager (S3) and click on its Preferences (choose from the "three dots" icon towards the right), then the settings will open in a new tab
- In Download History, select checkbox for the option
Switch to Downloads Tab at start of download and the changes will be saved automatically
- Close the tab to finish
The feature is otherwise disabled by default. By following above steps, when user initiates a download, the Download History will open in tab and switches immediately to the tab. User can also open on-demand using the keyboard shortcut.
Keyboard shortcut: Ctrl+J (default for Download History)
Download History: The add-on has its own download manager that is different from the built-in. The location bar will show moz-extension:// instead of about:Downloads for Firefox current built-in ones.
Works with Firefox 57 and later. Tested 5.12 with Firefox 70 on Linux.