With three games to play, including Saturday's crunch derby against the Cods at Blunder Park, United's destiny is in their own hands.
They trail League leaders Yeovil by a point, although Southend could reclaim the top spot - and knock United down to third - by beating Macclesfield tonight.
And Laws warned that nothing other than a top three finish will be acceptable.
He warned: "I've said to the players that we're in a position where the opposistion are playing each other. We've really got to push the boat out and secure the top three in the next three games, then go for the championship.
"It shows a terrific response from the players, and shows we've certainly got the right players in there. It's not easy and for the players to keep mentally strong and not wilt under the pressure is a credit to them."
To hear Laws' full thoughts on his first competitive return to Grimsby since his time as manager there, call the official clubcall line on 09040641003. You can hear a special phone-in, with United's supporters club chairman Andy Brown a guest, previewing the game on BBC Radio Humberside (95.9) FM tonight (Tuesday).