.SVG fonts change to default serif after publishing online with inDesign

I've downloaded and modified an Illustrator pie chart template I got from Adobe. I've exported the output to .svg and placed it in an inDesign document for online publishing. Everything works famously, accept the fonts convert to a default serif when viewed in certain browsers (Safari, for one), and look correct in other browsers. It does make sense that the vector graphics would be sensitive to a browser's constraints. 

None of this is traumatic - the graphic looks nice either way. I'm just wondering if there is a way to somehow control this for better consistency. I tried exporting the .ai to .png. The fonts then stick, of course, but the icon images are pixelated (as expected). Is there a way to have it all? 

Thank you.