Taylor, who once spent time on loan at Gresty Road, volleyed Cleveland Taylor's cross home at the far post to give Carlisle a 17th minute lead.
The visitors dominated for long spells largely because of the work on the ball of veteran midfielder Graham Kavanagh.
The former Stoke and Sunderland man unleashed a shot from virtually the halfway line as early as the eighth minute but Crewe keeper John Ruddy tipped the ball over.
Kavanagh's shooting was a continuous threat and he volleyed one effort onto the woodwork before Crewe emerged with John Brayford galloping forward to test former Alex keeper Ben Williams with a low drive.
Danny Woodards went close with a flying volley which took a deflection as it carried past the far post.
But it was Clayton Donaldson who levelled things up with a smart piece of opportunism at the far post after Icelandic loan midfielder Gylfi Sigurdsson flicked on a corner as the Crewe striker seized on the dropping ball and volleyed in.
Sigurdsson despatched a volley into Williams' hands as the half closed.
But Crewe, despite introducing Shaun Miller for Steven Schumacher at the restart, failed to gather real momentum.
Kavanagh was still pulling the strings in the middle and substitute Anyinsah was always a threat.
Anyinsah, another player who spent time on loan with Crewe, latched onto Peter Murphy's kick on following a Kavanagh free-kick and glided a header into the far corner for the 73rd minute winner.