Town saw off Crewe Alexandra 3-0 at Huish Park on Easter Monday to ease fears of the drop, and Way wants to see the winning feeling continue across the final three games.
Speaking to YTFC’s official YouTube channel, he said: “It’s important for supporters. You could see there was a sigh of relief on Monday. They just want to see their team win.
“That’s exactly what I’m striving for. It’s frustrating for me, asking why we had such a turn in form the January window but there’s a number of reasons.
“I want to make sure this club can get back to doing what we’ve done before but that’s going to take time.”
Way will go up against former Town manager Russell Slade at Blundell Park, having played under Slade during his tenure in Somerset.
With Grimsby rooted in mid-table following defeat at relegation threatened Cheltenham Town last weekend, the Yeovil boss is targeting another three points.
“Us as a team have got to make sure we strive to be successful and I want to beat the 50-point barrier and finish the games strongly with wins.
“They’ve got some good players and have got a wealth of experience.
“But every game in League Two is winnable and that’s one thing we have to make sure that we as a team believe in and go there and put on a performance.”
Yeovil fans can purchase tickets for the trip to Lincolnshire until 3pm on Friday from the Huish Park ticket office (01935 847888) or online. Tickets are also available on the day with a £2 matchday surcharge (excluding junior tickets).