Why does a Briggs and Stratton 4.5 HP lawn mower engine stop 2-3 seconds after it starts unless you continue to prime it?

If this happens to be a engine with a aquare paper air filer on it, about 4x6, behind the air filter box itself where it joins the carb, there is a funky shaped gasket there that eventually gets eaten up by fuel being pushed thru the primer channel, solution, replace it. This could also be a carb problem to though, question is, will it stay running if you continue to prime it? If it does, replace the gasket, if not do a carb kit, its easy. However...the gasket I am referring to, part # 272996, Briggs & stratton number, install 2 back on it, this will work better and last longer. If you buy a kit ( # 498260 ) do ask for a extra gasket, the kit only comes with one, but trust me, 2 works way better.. Your reason for even having to do this is state fuel, always use fuel stablizers, always ! Make sure when you put the air cleaner cover back on the 3 small screws holding the cover on, tighten them evenly, otherwise the will vibrate out.

The above answer is spot on. But, you can also check a couple other things. I just spent a day a half on mine searching for the 'simple problem' and in now runs like a top. I did replace the gasket behind the carb (as described above). But I still wasn't able to keep it running more than a few seconds. If you'll look inside the card while pushing the primer bulb, you should be able to see fuel pump up through the gold tube in the middle of the throttle body. If you see just a tiny but of gas or no gas at all, then try these additional fixes. The gasket for the fuel bowl must be in the groove and not weathered. The primer bulb must not be cracked at all. I still could not keep my running after all these fixes, so i turned the bolt on the bowl an additional 1/8 of a turn. It then started and has not given me any problems since. You CANNOT have any air leaks for the fuel to draw.

So here's your list that totals less than $20.00.

New plug, primer bulb, oil, fresh gas, new gaskets ( carb and fuel bowl). Tighten, but do not over tighten. This entire process is a 30 minute fix if you have all these parts on hand. Lowe's and Walmart will not have these parts. You will have to go to a lawn equipment shop where they usually keep all this in stock.