Gideon,
You might use Chrome inspector to see the HTML and CSS of the widget. Here what I'm getting:
<div class="ajax__tab_xp ajax__tab_container ajax__tab_default" id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs" style="visibility: visible;">
<div id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_header" class="ajax__tab_header">
<span id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_tab1_tab" class="ajax__tab_active"><span class="ajax__tab_outer"><span class="ajax__tab_inner"><a class="ajax__tab_tab" id="__tab_p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_tab1" href="#" style="text-decoration:none;"><span>Tab 1</span></a></span></span></span><span id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_tab2_tab"><span class="ajax__tab_outer"><span class="ajax__tab_inner"><a class="ajax__tab_tab" id="__tab_p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_tab2" href="#" style="text-decoration:none;"><span>Tab 2</span></a></span></span></span>
</div><div id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_body" class="ajax__tab_body ajax__scroll_auto" style="height:100%;display:block;">
<div id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_tab1" class="ajax__tab_panel" style="visibility: visible;">
Hi Gideon!
</div><div id="p_lt_ctl09_pageplaceholder_p_lt_ctl00_TabsLayout_tabs_tab2" class="ajax__tab_panel" style="display:none;visibility:hidden;">
</div>
</div>
</div>
So you should get something similar if you're using Kentico tab layout web part or a widget based on it.
I'd recommend you to wrap it with a div with particular id or css class and use it as a "prefix" for any classes and/or elements of the actual tab layout markup.