Chuks Aneke cancelled out Evan Horwood's early opener on the stroke of half-time before Ellis completed Crewe's comeback with a second-half tap-in.
Pools lost Andy Monkhouse to injury after eight minutes but his replacement Horwood took just 60 seconds to open the scoring. Jonathan Franks's cross was headed down by Antony Sweeney and Horwood lifted the ball over Crewe goalkeeper Alan Martin.
Crewe were handed a lifeline four minutes before the break when Peter Hartley blocked A-Jay Leitch-Smith's shot with his hands, but Mathias Pogba woefully blasted the spot-kick high into the stands.
Aneke levelled matters on the stroke of half-time, curling a shot into the top corner from the edge of the box, but the Alex went ahead when Luke Murphy's 70th-minute free-kick evaded everyone apart from Ellis, who touched the ball home at the far post.
Neil Austin almost grabbed Pools a share of the spoils, but his looping shot came back off the crossbar.